Transcript for:
Monotonicity and Extrema of Functions

Chapter 1 which is the function and application of derivative. Jaw Numbers and applications jaw. Application of religion jaw. Well, chapter 1 is functions and applications. derivative Okay? Then and I will today Now I will come to lesson 1 which is the first lesson. The first thing in the textbook is always the lesson. monotonous Now write lesson 1. Lesson 1 simple calculation monotonicity of function number. Ok. Lesson 1: Calculate the monotonicity of a function. The first of our class is teacher 1. theory You know monotony is what not? What is a single? Simply What? Monotonous yeah Now, single is a tone in the word form. tune What is monotony? A pose. A pose. Monotony is an abbreviation. common to both positive and negative variation. Homomorphic. Monotonous and inverse variation yeah Ok. There, monotony is a general word for monotonous and inverse variation Simply One, the tone is the shape. Monotony is a shape. So a shape if it increase is to go up is what people call copper variable. The symbol is the letter increase. Ok. And the symbol is an arrow going go up. Remember? What is a paradox? Nhai To be reduce. The symbol is an arrow going down. ok So you have to understand monotonous teacher is writing for the common people variable and inverse Okay? Copper the variable people call increase and arrow go go up. Inversion is to decrease the arrow. down. Okay? Well the theory is as follows. Given the function y = f(x) with derivative on k, there is derivative has derivative have religion upper jaw on about k y = fx has a function segment on the interval say if this x1 is less than x2 there but fx1 is smaller fx2 here x1 is smaller than x2 but fx4 is smaller than fx2 How do you see its signs? same sign right x1 less x2 fx1 fx2 again right? It's the same, same sign is called homogenous Monotonically increasing function. Homomorphic homogeneity Can you abbreviate? Which when when then my derivative will always be greater equals 0 for all x in interval k. Okay Not yet? There. When the function is increasing Not yet? There, monotonous means it has the same sign. This. X1 is less than x2, fx1 is less than fx2, correct Are not? What is same in Chinese? To be copper right? There, the same variation. change. X1 is smaller than X2 f(x1) is smaller FX2. The meaning of the sign is the same. Okay? There. Then f' will be positive You know what you learn. grade 11 What is F'? Is the correct slope? Are not? Positive slope. What does it mean? When it increases this way, when the function is increasing this will arrow up. But go up so what? go up is me is more and more bigger right? Growing up must add to add to it must be positive yet? That is the same as the variation. will be positive going up and then more and more increase is more and more increase will be positive In this second case, if x1 is less than x2 fx1 is bigger than fx2. See two Are these signs reversed? Ah the opposite what is the reverse Is it naughty? Reverse right? then go against variable. Paradox. Paradox. Already and when that function is decreasing, then the you know what? So what? Then f'x mine will always be less than 0 x x belong to k. Already yeah So, to conclude, the increasing function is What is the sign of my f', guys? positive sign there. Inversely, our f' carries sign minus. Ok. There. So the relationship of derivative What is the difference between the two? There. Positive and negative are opposites. See? Such This yeah It's not entirely accurate. It one more thing guys big f'x equals 0. Is this monotonically correct? f'x less than 0 and x belongs to k is inverse right? But guys, it's just zero at some points only. Why so? What if f' is always 0? Then What is the function, guys? f' is always 0 so what? What is the function that is always equal to 0? Thing What function is always equal to 0? What function? is always 0? What is always derivative? equal to 0? Right. The constant y = c is a constant. number. If the number is constant, this is called an unchanged function. yeah It will go like this guys yeah. It will go sideways like this. There. Then You have to add this for me. What is an additional condition? Equal sign only happens at a few points only. Take note teacher The equal sign occurs only at one some points Remember me Equal sign by just happening here. Equals sign here only occurs at a few points. Where is it? What is always 0? is a constant gone Here you go guys. This. there teacher let me draw for you guys Ox, oy this. Ox, oy here This is the line y = c here Is a constant that does not have to be c then the derivative is always 0, right? There. So if the derivative is always equal to 0 is called constant. Still Where the copper is closed and the reverse is the meaning, it is only zero at some points only. Only happen out at some point. In some point. Remember this sentence, teacher. Sentence This exam is very good for you I like to do theory. Then why? Because f' equals 0 then function is constant function right? Always 0 is a constant function. Then let it be. accurate and unambiguous then f' me will always be equal to 0 for m x in k. There nh is equal to 0 k. Here I will take you one the example is as follows. Look at this example look like this Now, this is what people call a variable table. this heaven guys Variation table This. This x, the new relationship between this x, y' and y. Well, let's say people give it like this. yeah This is very easy for you to get wrong. this mix This X for example is negative infinity. yeah This is 0. Here are 2 go. This go on Ok. There. This is 0. This is 0. ok For example, this is a sign positive. Positive sign. positive sign for example. This is what people call a variable table. oh my god The relationship between x, y' and y called variation table. Relationship and guys are you positive yet? positive equals 0. So you see your y' how? Always greater than 0 or the same Is there a zero sign? Then the graph what will it be? Homomorphic. And the I have to write a letter to the teacher up like this There, the function is increasing when positive or negative is equal to 0 at a few points Understand? No way. write like this This ocean goes up. Happiness go up here. Happiness goes up like this wrong. Okay? There, remember that picture. hey Positive zero positive zero negative. Oh there all positive and equal to 0 then the graph will go up. Remember? There, write it like that. always Remember? Similarly if the sound The sound is written down immediately. Remember, Remember how to identify the easily confused children. This. There, or I'll draw another one. yeah X this y' and y. There, for example this is negative sound. same This is 0. Very positive This. There. Teacher, teacher, this is number 1. Go. V here is this 0. This is the sound minus. Any. You guys graph the shape what is it like How are my siblings? So the sound is go down? Sound is inversion. This guys Look at this. If y' is negative then the function What will the number be? Paradoxical right? What kind of arrow is the reversal? Go down. There. So look at this. Sound go down there Zero is also a denominator. always. Is it ok? Negative or zero is the opposite. So the arrows all go down like This. Do you understand? There. This is something that you can easily confuse. Much you are negative i go down to 0 no more so this sound I go again. Understand? Is it wrong? That's why This is very confusing so this teacher must emphasize carefully It is just 0 at a few points Stop. Ok. Then here you supplement the expression. I am with the sample x belongs to the students for this teacher Here f' = 0 for the sample x belongs to k is a constant function. There. So with m x k ​​means always equal to 0. Do you understand? F' must always be moral If the function is equal to 0, then it is a constant function. equal to 0. There are some other functions that are still derivatives. equals 0. But remember that f' is a function. equal to 0 then it is a constant function. Okay? The nature of homoscedasticity what do we need? Consider the sign of y'. Remember me to close one. Okay. Now write it down for the teacher to decide. Which one? Okay. The essence of monotony that is the sign of y'. Ok. So calculate single monotone tone by accent miss you teacher Close monotony is the sign just y' Okay? That's it guys miss. Now you listen to simple adjectives. harmonic or homogenous variation The quality is to consider the sign y' y' is positive is going up homophone sound d sound correct no this sound one the negative person is the one who will go down no positive is more and more go up in the air and then to the teacher This one form one gives the variable table given table of variables Variation form 1 given variation table. What is a variation table? Remember The variation table is the table that has x, has y' and have y. Okay? There are three of those called is the variation table. Then look at the teacher's example. 1 like after. What is example 1? Given the function f(x) definite and continuous on infinity positive infinity has the following variation table: following image. That variation table is the key. the relationship between this x, y' and y. And from here You can see a lot of things. The first you see the mark of y' here. Positive should i go up Yang goes up to Yin stab down Okay? Duong then go up. There. Then we ask the proposition Which is correct? Just look here and conclusion 5 seconds. What is the inverse function? variable on the range 1 to positive infinity. True or false? 1 to infinity Children this observation 1 to infinity then the function its number goes up. Go up or down? Go what should it be? Homomorphic. Is the reversal wrong? wrong yeah 1 to 10 positive wires going up. There increasing function on the interval from negative to infinity together to -2 true or false? Look at this. Minus this infinite -2 where are you guys? A -2 where are you guys? -2 must come before -1? Less than -1. Then -2 must be here. Are not? Is that right guys? Yes this is -1 no? If -2 is less than -1 then it must be front charge -1? There. So then guys look at the pronunciation for me go to -2 Is it positive? Or go what do people call it? Positive what is going up Correct covariate Are not? There. The graph of y' is positive and goes up. Ok. True or false, kids? Negative to -2 graph goes up -2 of you guys come down here yeah There. Up yet? Ok. That's right. Yes. That's it. Where is sentence C? why wrong? analyze with teacher Function inversion on the interval from negative infinity to 1. Look at this. Infinity to 1 What does the graph look like? Sound from infinity to 1 Does it go up and down? Then can not be called a reversal. What is paradox? Just go down right? Up and down not called a mutation. Function monotonically varying from -1 to zero together. Come on guys. -1 to the end Are not? -1 to this end. Look at the graph its Up and down? Right? Prove that it will not be variable There, the same variation is just going. up only, reverse is only down. Remember? Monotonous. Ok. One is B. Come in. This first talk. I want The whole class recognized the transformation board. what is its nature And look at Here, circle the answer. Come on, shit. credit to the teacher Given the function y = fx, The mean square table is as follows. Function y = fx on which interval is the inversion? Paradox is going down right guys? Then or There are two ways to recognize the sound y'. There You guys rely on y' sound or rely on y also Okay. On what range? -3 to 1 kids this observation -3 where? -3 in here before -2 but less than -2 where is 1? 1 here. Oh look this is definitely wrong Already. From -3 to 1 it is both positive and negative. positive right? It's messy. So wrong The only way to go is to go come down What about 0 to positive infinity? Look from 0 to positive infinity? Go graph up? Then it is monomorphic. Paradox wrong? Then wrong right? Infinite Sound to -2. Negative infinity to -2 graph y up wrong right? The paradox is me must go down Infinity to -2 The graph is not going down. Wrong. D is what is this? long -1 where? Know where -1 is? -1 it must be greater than -2 and less than 0 Are not? So -1 is at this position. Okay Not yet? Then you look at me from -1 to 0 This. -1 here I line down this. this 0 is this right? Then look at this. From -1 to 0 graph going down? Down is it a reversal? Exactly there. Always circle the time to knit D. They glanced play and remember theory only A few The first form I give you to remember theory The arrow is going down and y' negative. The arrow is the direction. up and y' positive. So what is example 3 guys? For function The number y = fx has a variation table. which drawing? Here is the signboard. Jaw The number is decreasing on which of the following intervals? Reverse is down. ok Or is the negative y' of the two ways of knowing. what sound Does infinity to 2 have an inversion? Minus infinity to 2. Look at this. Sound let's go to 2 This sound is infinitely 2. Minus all negative yet? You see both go down right? So guys this one right or not wrong? You are right, guys. You know why not You read this for me. Look at this theory. Derivative above k. To have a derivative, the function must be related. customary Right guys? Continuous function then new derivative Then you guys look at it. Teacher, this question 3. From negative infinity to 2 even though it is negative right? But it not continuous Remember it was cut into pieces. two parts? Cut in half? Then it will not be able to co-vary on about there Its own space what should it be? It must be continuous. Children wrong because not continuous. No connection customary No connection customary Okay? Missing teacher, just monotonous is a stroke, it must go down like this is always not down one this release Have to go down one more time not counted Remember? One time one always play Which 0 to 2 is ok? Are not? Look at that. Look from 0 to 2 go down one shot? Right? Look from 0 to 2 here. From 0 to 2. Graph down yet? Exactly. B there There. Remember that. This is very easy to get wrong. But it has to be continuous. Okay? It must be continuous to calculate the coin change change Which is sentence C? Two input lines. together. Two infinite lines are going up. just the same False or positive zero together. Not positive just down just up not count right? Not the same, not the opposite. Here, remember this theory teacher. there you guys can rely on the arrow to go up or down arrow Okay? Or what can you rely on? remember? The f' Yang is copper, yin is inverse. Okay? So this lesson guys no arrows right? If we don't see any arrows, then we based on f' only. Which is increasing then f' must be positive from -2 to 0. Where is the -2? -2 This. -2 to 0. Oh, you see the y' Are not? Is it both positive and negative? Wrong Already. Are not Okay. The function is decreasing from infinity to 0. ah Oh, negative infinity to zero. Oh my both negative and positive No way. Are not Okay. This sentence c is a decreasing function on from 0 to 2, you see from 0 to 2 this 0 to 2 graph f' negative yet? Sound is opposite gone? Okay. Ok. What is the answer? To be do you remember? That sound is negative. Copper negative Ok. The above decreasing function range from minus v to -2. The sound v goes up to -2. Oh my god. Yang then How can the inverse be achieved? It must is it monotonic? Wrong. Next sentence 5. Come on question 5, please choose for me to play. what is this This example 5 of fx has The derivative table has a derivative sign table. as follows. Which of the following statements is true? The function is increasing on the interval -3 to 0. -3 to negative 0 must be detrended. Wrong decreasing function on the interval from 0 to 3. 0 to 3 negative inversion yet? Correct Already. Đ is B. The inversion is correct. What? Right? Already. That's the first form. the form that the teacher lets you recognize Stop. Look at it and circle it. Then I will change to form two for teacher. The second form is You have to make a Minn table to give in one form. Not given. Form two this Form two. Create your own variable table heaven Self-made side table heaven Come on you have to be independent yeah Step 1. Step 1. Step 1 This. Step one is the class we will have to find. practice for teacher The domain is find that condition Practice identifying. Find condition. The second step of our class calculate y'. Have you calculated y' yet? Already search point find point x0. You have to find it teacher x0 do for y' = 0 or y' is not true sure Find points makes y' = 0 or y' undefined. Unspecified means other sample 0 that Do you understand? There. Y' no identify. If not, I will be two bricks Not sure if I will hit colon yeah Then step number three. This third step is for you to be independent. variation table for teacher The three-letter word chart is: There are also three lines like that. X, y' and Y. There, x, y' and y. Okay? There, set up. This. There, I will fill in the values. x0 makes the function y' = 0 and 0 is defined that And you have to arrange them for me in order. increase gradually Ascending order. Ascending gradually Arrange in ascending order for the teacher. Ascending gradually Then step number four is conclusion Stop. Conclude discussion Step four is the conclusion. Look here then surely i know the class right Are not? It divides the first form teacher for them Then conclude. And you guys remember, you have to use words and give teacher Must use words and must use words and should I use the word and? must use from and to connect spaces can not be used can not be used use mnemonics? Not to be used fit. Whoever uses "suit" is wrong or Don't use minus sign like this either. no That is not the word Is it ok? Do not use in combination I have to use the word and for the teacher. From and that must use words and sound Copper function variable on this range and that range this range does not match that range is wrong. Okay? Required words and for you Then copy the teacher four steps enter. Step 1, step two calculate y' 3 is the table heaven Step 4 Conclusion yeah I will use those four steps to create variation table Here you see this six example. The teacher guides the students to set up What is the variation table of this? Consider monotonicity of the following functions. Jaw This third degree, this quadratic function. What is this function? The best of the best best. What function does this function do? Second degree above best. What function does this function do? The square root function Are not? There. I will guide you to set up these five functions. And these five functions are five functions. very common Everyone must know. And later you must know the shape of these functions. Come on! First I will come to the first function. Level three. Present carefully and in notebooks for the teacher. always self-study yeah This doesn't have to be done quickly. do nothing Do it well. This will be later also applies to speeding up other things. Come on teacher question A yeah Then step one what do you do? Step one find what? Correct set Are not? The domain of the cubic function is What? Have you had any wine? No conditions at all. Which is the third level of the corpse? define d = r. Right? Second step calculation What? What is y'? Calculate the derivative. Y' equals some? 3x minus how many? 6x plus what? It's over. Already You guys will give y' equal to 0 for me. In because this article has no points for y' is not specified. Right? Let it be 0. You guys just give it a 0. Already From there I can calculate x? What X? x = 0 and x = 2. Ok. Is this ok? Then after that I will do the third step. sea ​​board yeah X this y' and yes Okay Not yet? Already. Come on guys. This x guys You have to arrange the following in ascending order for me. no? You see the special features what is my difference The domain is what Where is R from? Infinite Sound to infinity right? Then sound together to the positive and negative together. Then you fill in 0 and 2 Arrange from smallest to largest. Correct Not yet? What do these two values ​​do? equal to 0 guys? What do 0 and 2 stand for? equal to 0? Make y' equal to 0 right? Right? 0 and 2 here. Make y equal 0 yet? Then you fill in two boxes. 0 in This. Then it looks difficult to set up but the If you do it step by step, it will be easy. Step Tuesday after we set this up You have to look at the sign to see if it goes up or down. can i go down How to check the sign? Opposite to with quadratic function, quadratic function then whole class Can you remember what it is? In remember the left one? I am consider the y' sign. y' is a exponential function. Listen to the sentence in the outermost fruit yet? Not heard This sentence is learned in grade 10. In this fruit is 3 positive or negative? Positive. In two this number 0 In between these two zeros. Left mark this number This is positive. In is left is negative. Remember? The outermost is positive. Do you understand this? There. 3 positive in the outer left. Okay? There. What about Duong? Yang goes up right? What about sound? Downward ok Remember to draw this correctly. yeah Okay then, look here, kids. concluded that the above function is increasing what time and what time not? Then end discuss with teacher The above function is increasing. Copper Function interval What range is the variable on? On which interval is it monotonous? On the range from negative infinity to negative nothingness to 0 yet? Is the graph going up yet? Minus tickle what else guys yeah? And what? And from 2 to the same graph up yet? Then 2 come in together. Ok. Then the function decreases approximately any? YES. Inverse function inverse variable right? On what This? On 0 to 2 the graph goes down Not yet? Then 0 to 2. Ok. Remember that, kids. teacher use words and pronounce From and Not yet? That's it. There I look at the rock I just circle it. Is that it, teacher? Go. Ok yet? You have understood the steps. Not yet? Just follow those steps to take the test. close to the function Here is sentence B. Equals function how many here Level 4 right? what teacher quadratic function what is y y' is a set of calculations. determine in advance What is a domain? remember? What are you practicing identifying? What conditions determine the quadratic function? Are not? No right? Then d = R. The next step is to calculate y' for the teacher by some? y' equals some? 4x hats 3 minus what? - 4x is 0. Let it be 0 always Right? What is the equivalent? This? 4x smaller. X squared minus 1 = 0. So what is x? x = 0 x = -1 which is correct? I calculate x = 0, x = -1 yet? Then I will make a table sea ​​of ​​heaven Stop. This X y' and yes Ok yet? 0 and plus -1. This is the sound this infinite -1 this -1 is less than 0. Until this 1. Similar together. These values ​​all make y' = 0 out. Right? These values ​​do for y' = 0 right guys you? Right? This is -101. Children calculate y' = 0 this time using calculator is too simple simple Right? Already. Then come How does our class check the marks here? huh guys? You know I consider signs like which or not? What is the sign like? which one? Then the teacher guides you how. Examine the signs of basic cubic and quadratic functions as follows very quickly Normal I can use a computer but how many? You must know this basic function. what is it guys? You will be teacher This. What is the sign of y'? Here here positive or negative? 4 positive right? Female here is the sign of y' guys. For example, this is a sign what is this? These 3 are positive or negative, what is the difference? The outermost mark is always The largest coefficient of y' has the same sign. Not yet? Finally, remember me. outermost outermost Outside together outside and then knit The sign applied to the basic functions is cubic and quadratic functions. Remember? Where are those? more complex functions, the teacher will guide you guys how to press computer Okay? But this article takes longer to type on the computer. outermost and interlocking. What is the coefficient y'? positive or negative? Positive. So the outside will have the same sign as positive. Is this outer space okay? The outermost sign is positive, so what is the direction? This? Yin continues yang continues yin. Knit sign Not yet? understand? In addition to the sign, we must consider the sign of y' but. This is the outermost here. Sign of y' is the number here is number 4 then outside will be is it positive? Then the children were enamored for teacher Okay? The outermost is knitting. this outside mark the outside space will be same sign Same sign outside with same sign inside of the sign of a is the outermost sign of a. Understand? Together that's the a sign Is this 4 okay? Same sign same sign Okay? Already. So then you look here you know its direction yet? What about sound? Down positive Yin down yang up Remember to answer me directly. Already So you see the conclusion function. What is the function? The above function is increasing. about remember? Monotonous on interval any? Above any? On which interval is the function monotonous? Can you knit the stamp yet? What is a stamp? Is positive turns to negative, negative turns to positive, positive turns negative. Remember Not yet? There, I will guide you later. he clicked on the computer But this article is actually It takes longer to type on a computer than to do it by hand. so we have to know both to choose one fastest Let's change the interval variable. Hey? See that positive space? This interval is not and this interval is not -1 arrive 0 and what remember? And what? Is -1 to 0 positive? 1 to infinitely positive Not yet? Ok. Already. The above decreasing function what time On what This? On what range do you remember? Above go away down Infinite Sound to -1 and what? 0 to 1 yet? There. Already write it The negative infinity is 1. and 0 to 1. There. ok There. The steps I took Are these functions ok? Then the you fix it So what function is C? Level best of the best right? This sentence What is the condition? Here's sentence C. Thing this determination. The domain is what do you remember? The sample must be different from zero. Are not? As x is different in general is me Find x other than -2. That student is x other than -2 d = r - 2 Then continue this. Finished calculating for this teacher. my class remember the quick calculation formula of the guy this? Which formula is better? If anyone forgets, you can write it down for me. what is the formula teacher ax + by + b What is the derivative on cx + d? By what? a x d minus c nh b is not divisible for sample bottle not right? So that's it then what am i? what time is this? A cross multiply a times this is 4 this 4 Is this 4 correct? Subtract this cross multiply how much 1 times -1 is -1. What is 4 - 1? 5. Ok. So have you calculated the result 5 yet? Mentally in mind 5 on x + 2 all jar. Okay? 5/x + 2 body. And you observe this. 5 positive yet? Do I have to check the sign of y'? 5 positive yet? Don't give it as 0. Zero there is no experience right? There. 5 positive sample positive yet? Ok. So always positive Already. Always positive right? So we don't need to solve for zero anymore yeah What is the point of solving for 0? Always positive how can it be 0 So the table teacher x this, y' and y This. ok huh? X Y'y Y guys. my x is what is this? X is different from -2 right? Come on, sound! this same, this infinite positive. Right? You have to fill in the values ​​for me. make it indeterminate. At point -2 is not determined yet guys? Then You guys fill in teacher -2 here. -2 no how to determine Just two bricks. At -2 makes y and y' both undefined. determined You give me two lines down below This. ok There. Give me two lines yeah Two dashes represent an undefined property. determined Okay? Now let's consider How is your mark? my y' always What? Always what? Always positive? So for teacher two plus signs in not knitting mark y' is always positive, right? What if two plus signs are positive? yeah? Go go up go up note That's the first table. it looks like this because it is blocked way at point -2 understand? It doesn't determined at point -2. And then what will you conclude? so guys? This conclusion is very important. this important What is the conclusion? you? What is the conclusion? right? Conclusion as any? Conclude yeah Conclude hey guys What is your conclusion? Conclude this thesis. Jaw Functions with only inverses and only co-inverses just change right? Homomorphic. Copper variable on what This? On negative infinity to -2. And remember I use the word and. And -2 to the positive together. Okay? The class I remember for teacher use words and sound And is it ok? Jaw increasing numbers from negative infinity to -2 and -2 to infinity. Remember? There. What do people give for sentence D? Jaw y = x² - 2x + 5/ x - 1 right? And How do you calculate the derivative of this function? remember? Don't remember the formula? U/V derivative equals What? By what? U' times v? Martyrdom function multiplication denominator minus denominator derivative multiplication what right? What is the division of factors? Sample is that correct? There. Who forgot the formula? Remember to calculate the derivative for the teacher. then What is grade 11 related to grade 12? Application use derivative There. So first What is the most common plus sign? you? How different is X? Y other than 1. Then next or our class is more accurate than you guys What should I count? D equals R - 1. But I'll just write it like this. familiar Let me calculate for you Go. This guy has a feature formula. yeah But the teacher advised the students to study. what do you do if you know the formula It also harder than normal Calculate for teacher in normal way for fast U' What is the derivative? 2x minus 2 yet? What is this u' times v minus x - 1 v'? V' is teacher what is a? 1 person Not yet? Divide some? Sample jar. Ok. Guys, let me withdraw. let's go Now shorten the teacher shorten it quickly guys The teacher shortened it. sample form teacher can recite quickly Teacher think like this. What is this 2x times x? remember? 2x² minus x squared gives 1x². Already continue this. x coefficient 2x times -1 is -2x. Subtract 2x more to get -4x. -4x Add 2x to get -2x. Understand? There. Multiply these two together and you get -2 times -1 is 2. Not 2 but minus 5 is to -3. Right? B can be written as a division sample vase Ok. Then the whole class finished calculating y' Not yet? Then I will find the points makes y' = 0 and y' undefined. Then you know y' is undefined at which point? Y = 1 is the condition just calculated right? Then y' = 0 go guys me let y' equal 0 So you get x equal some? how many guys? What X? In here without pressing the machine calculate for me Oh, I will mentally calculate x. equals -1 or x will equal -1 or what? Or by 3. Do you think it's right? -1 3. Okay After that, you make a variation table for me. Go. Let's discuss which heaven Then I will create a variable table. oh my god Now, let's establish a body for the teacher as follows: any. X this y' and y This. Is this right? You look at the teacher as follows. X will be from negative infinity to positive infinity right? That's a great sound. very positive Okay? then y' what is mine y' guys look at this You have to fill in the prices for me. value that makes y' = 0. Here it is. Look at reason theory here Find the point x0 that makes y' = 0 or undefined. Okay? Already I filled in the order of increasing prediction. Okay yet? Teacher fill this out. Ascending which one? The first one makes him uncertain. set 1 here -1 smaller than the front 3 larger more behind. Right? Then you guys look at this One guy one guy in the model do y' is not determined, y is also not determined always. So guy 1 is worthy because of two reasons. brick. Ok. What about -1 and 3? Make what is y' Is y' equal to 0? There. Teacher rewrite here Right? Do for 2 teachers 0. Okay? Already. Then now teacher need to check the sign Oh, you know how to check the sign like which? Then you look at me like this. The teacher's sample is always positive. yet? You see the sample is always positive. Are not? The sample is always positive. Ok. Teacher left sample teacher The sample is always positive. So the sample oil test is not important anymore because it is positive So what is the degree of the numerator? What about level two? What is the formula? Inside the outermost fruit. In the fruit outermost remember. Second degree is in leftmost Then you look at the two the number 0? Two zeros yet? In The opposite of two zeros will be opposite in sign. What sign is this? Is the square of X positive? negative sign? Positive. In the left. So two this circle There are two in the left. minus. Okay? Besides the two marks round is the outermost positive sign that has understand? I will hide it like that. Quadratic function auto is inside left outside together yet? That's like in two numbers. 0 is opposite sign, outside is same sign. Ok. There. Or you can try the value here There is another way for those who can not do it quickly like this if you can not count then guys You can substitute any value here. Wallet For example, let's substitute x = 4 here. For example You choose a value of 4. From 3 to Tonight, you choose number 4. You press your computer instead If it will definitely have a positive sign. understand Are not? Or our class can be from 1 to 3 students replace teacher number 2 in the middle understand? Replace number 2 here and press it yourself know change? See which sign fill in there Okay? But the article hey i told you guys, quadratic function remember the teacher is in the heart and the same to I will do the test later. Difficult lesson more in the left and the same mark goes with the me What about Duong? Go up. What about sound? Down goes negative down positive goes up. Ok finished conclusion Go. So this. So teacher, please conclude here. So This. So the function is increasing on which interval? Hey? Infinity to -1 and 4 to and 3 to infinity yet? This is from 3 to zero. together? Infinity to -1. Function copper variable on The interval above the interval is from minus infinity to -1 and several and 3 to the middle of infinity. There. Continue. The above decreasing function about. This paradox what on top Inverse function above about -1 to 1 and 1 to 3 but not get one is -1 to 3 is wrong ok? Because -1 to 3 is nothing. guys Not continuous. It was two brick, is it separated? See, no. The conclusion is -1 to 3 but must be concluded The teacher's argument is on the range -1 to 1 and 1 to 3. Remember? No conclusion is from -1 to 3 is not wrong. In because it is cut off in the middle. Okay Not yet? The two bricks are what you have to separate. take it apart piece by piece. Then you copy the theory in. Copy post in Come on, big teachers, sentence E like this: after. y = square root of 4x - 2x Are not? What is the condition for the square root function? What to do with a radical expression? greater than or equal to 0. So e condition 4x minus 2x² must be greater than zero. Take the calculator and type it for me. I know how to press right? Get your computer and click. Degree function I know how to press. Click any teacher equation go This is too much enough Click for teacher to see x like any. Teacher, please count. The teacher mentally said as follows: This. Will it have solutions of 0 and 0? and 2? where? You can taste the teacher. 0 and 2 is 0 and 2 but have opposite signs. So it will is from 0 to 2. The kids looked at the computer like this. Are not? Ok. Th x lies between 0 and 2 correctly Are not? Ok. Calculate the derivative for me. Yes some? how about you guys? Which root is u. what is the function? Derivative root equals What? By what? U' on 2 square roots of u no? You guys go derivative. how many This? Which U? This is u This. How much is peach? 4x derivative equals u' always this 4 Subtract uh -2x squared derivative is the head that falls off the hat is 4x. What is the sample size? The sample is What are these 2 units? 2 square root of 4 x minus 2x². ok Okay, okay. Ok. What is the result? give me 0 How much is my medicine? I calculate y' and then I set it to 0. Okay? What is 0? What X? X1. Does X 1 satisfy the condition? Have satisfied right? X 1 must satisfy Is the condition from 0 to 2? Standard Not yet? Already. After we have x = 1 Okay, x is from 0 to 2. So guys will create a variation table for the teacher as follows. Which is the dock, guys? X this, y' and yes Please fill in the values ​​of x for me. any. Where does your teacher's X point from? okay guys? This condition, x of teacher from what to what From 0 to 2 only right? Right. Then I just x words 0 to 2 Stop. Okay? That's 02. Then back to If x = 1, what does y' equal? Already x = 1 then we have to make y' = 0. right? Right guys? Then x is 0 Just take 0 to 2, guys, don't take it. add anything No need to should take negative infinity or positive infinity do nothing Which interval does X belong to? I give you exactly that amount. Any Now how to judge the sign? Consider the sign how now guys? Guess the kids How to check the card mark now? The teacher remember The teacher guides the whole class. by clicking this calculator. Which friend? don't know don't know how to see the sign then the whole class chooses any value for teacher in the middle 1 in 1 with 2 then in the middle what number 1.5 is not correct? 1.5 no, you guys replace it here for me see if positive or not Take instead This sample is always positive. yeah Get rid of this. Get a teacher for a price Any value is 1.5. Here are my students. Any take the calculator and click. I recite to you by computer click this. 4 minus 4 x 1.5 negative or positive? 4 x 1.5 = 6. 4 - 6 must be -2 which is negative yes Is that right? You choose a price any alternative here or not? Too easy Just like a child. Come on 0 1 choose any value choose what? 0.5 right? 0.5 0.5 how much is it here? 4 x 0.5 is 2. 4 - 2 what? Is 2. So the result is positive or negative? Hello guys. Exactly always there. Or who doesn't like to click Computers can make sure of free will. I can remember what the teacher told me. Outermost What is the number with the coefficient sign before x? This? The x coefficient is negative, right? So Is the outermost sign negative? Then knit sign yet? The outermost stitch is also fine. Okay, this is easy. go go up and down then give me the result Go. What is the conclusion? Go ahead. The function is equal to the interval multiplier. any? On what range guys? Copper variable from 0 to 1 Not yet? Decreasing function on interval any? Where are you guys? Above 7 to what? 1 to 2. Ok. Finished multiplying the numbers. Then you conclude the lesson. The articles do the same So we have just Five sentences correspond to five forms and five functions common number 3rd degree, 4th degree fraction first degree above first degree second degree above second degree first and square root. Okay? Remember how to do steps Is that ok? Go There will be special functions as follows. Children just look This sentence 7. Given the function y = x m 5 + 3x m 3 minus 8. Right? Exercise What is determined? When to find practice Determine for the teacher first. What is D? d = shake? Because there are no conditions both right? Continue calculating y' for the teacher any. Y' some? 5x hat 4 plus 9x squared right? 8 is equal to 0. Okay? Do you notice anything? The power of 4 is always 0, right? Is the square greater than zero? Hat even will always be positive. Okay? So evidence what is your y' here? As big as 0. X reduces R. But y' is always greater than 0 what is the function? Look at the theory This. Is Y greater than 0? f' is greater than 0 then what is the function always like guys? Copper gone? Homomorphism on the corpus its always determined So the function is homogenous. just change on R. y' is always positive. copper variable on R d equals 3 no need to make first table where So for example 8 in y = x + 2/x - Which conclusion is correct? Not yet? First you have to find the corpse. let me know What is a domain? remember? Other x condition some? What is the difference between condition x? X is different from 1. I agree that the condition x is different from 1 is true. Are not? The second step calculates y'. y' equals some? how many girls Climb down minus climb up -1 minus 2 is some? -3. Can you count? There. Divide by sample down cross product minus up cross product right? Cross product down minus product cross up. Then just get the coefficient. Already guys What is sound? always negative Are not? Positive sample. So prove this always what? Always negative, right? Always negative. With sample x belonging to D. Does Vx belong to D? Right? Have must always be negative for x in D? So then what does that prove guys? The first look guys, we can't take this sign Do you agree? in the teacher's note here. Here you guys look at this caption. Right use and do not use or is this minus sign. Right? Obligatory must use words and pronounce Ok. Then sure this sentence is definitely wrong Wrong. This sentence D wrong yet? Its condition is x other 1. Then definitely not possible inverse on R is possible. Single mutation on R Okay. cross it out right guys? Because Its domain is just m different from x If it is different from 1, how can it be agreed on R? right? It thinks this guys, it will like this you can write quickly for me This. 1 this x different from 1 then you cross it out right? That and always negative always negative. Write this is quick So the inverse function Has the variable on negative infinity gone to 1? And 1 to positive infinity because the negative sign is correct Not yet? Then knit yourself as C always. That's it! just show like this not to make a comment save time There, how to set it up? then the teacher just said above right? Where is it when I do my test? should maximize time right? I do it faster. Draw the axis for the teacher this is ok This is called the axis of sign. Stop. No need to use table change Given the function y = square root of x - 2 right? The above function is increasing. how far from here First you find conditions for teachers The condition is What? What are the conditions? x minus 2 must be greater than zero and x must be is it as big as 2? So then draw your own class calculate y for me how much How much This? y' will be equal to the derivative of x - 2. 1 above then here are 2 roots of u'/2 roots of u correct Are not? And you guys look at this 1 always? Is a positive root always positive? So always positive? Always positive? So it will always monotonous You have to draw for me just a short axis Well, is it? Is the condition x greater than 2? x this is 2 times bigger. So you delete the small part more than 2 go Right? Already. And my y' what is this? Always positive right? So then it is definitely positive. So it will always what guys? Always be positive up? Always monotonous? There. So what is knitting for yourself? Here on R is wrong. How on R? Accumulate it from two to positive infinity how can R be right? What is wrong? 0 then can i use it together? No 0 guys. 0 cannot be 0 in here X is greater than 2. So 0 does why is it greater than 2 right guys? you? Wrong? Wrong. The infinity sound is 2 wrong notes. Because x greater than 2 and x is greater than 2 then do How can the sound v go to 2? Unreasonable. You have to see its condition. what else It only varies on the interval. from 2 to positive infinity only. So 3 comes can infinity be obtained? Have. Then 3 where? 3 here guys. 3 here 3 big more than 2? So you take from 3 to positive infinity is it still positive? Still Is positive still monotonous? Exactly. D is just that but not necessarily do you understand me? But that is what i present for it standard textbooks for children was forced by the teacher to do practical work. Any Sentence 10 This. Given the number y = fx with derivative on r' right? This sentence is given by people. I have derivative on R which is the domain Already. Is the domain R? And people f' has been calculated for you. Here you go guys look at the theory for me Which is Normally I have to find the domain Are not? Is the lesson given? Already Has anyone calculated x for you? So what is your job? Stop? Find the point that makes y' = 0 or y' not sure right? Then I set up Is the board durable? People do ready for you to practice identifying Calculated Already. Just let y' = 0. Okay? Find the point where y' = 0. Because it belongs to R then what to do if you can not determine correctly Are not? Belonging to R is definitely determined. So I will continue with the next step. will give y' = 0. f'x = 0 what do we get? Now, let f'x equal 0, then each equals 01 right? Let this be 0 then this guy is 0 teacher is 0 or that guy this is 0 right? Right? In because each one is equal to 01 and the product is there. So what is x, guys? Head x = 0 x equals what? f(x) = 0. The first equal to what 1 this Next one add -1. What next? 2/3 right? Then this is calculated mentally. Anyone can calculate right? Then continue Next plus sign teacher make table correct variation Are not? X this, this and this. Okay? Now you fill this out. This article it determined on R from negative infinity to infinite wire together. Right? Then you fill in the price here. cure A -1 this 2/3 and what? 1 then makes y' = 0. Is this correct? There. So now you consider the sign like that any? Now how do you judge the sign? When it is a product Consider the product form then how do we do it How do you hide it? Now it no way you guys give example on the left the end or whatever. Because of this There are no accent marks, right? you? For the product form, in our class we should click calculator for teacher This form of product product form product form of f' ok? Then you just need to press Casio teacher as follows. Casio will definitely come out. yeah For the product form, you use CCO. Any You use the can key for the teacher. Does anyone know how to use the can key? Head First you type f' into the calculator Go. You enter f' and the calculator for me. come on Teacher, please guide us to click. as follows. First you enter the f' right? f' right? slimy kids this teacher 1 - x times x b - 1 times 2 - x 3x. You guys enter it into the computer. 1 - x this times x squared - 1 times 2 - 3x. Oh guys Have you entered this into your computer? Enter f' nha. You will enter f' for me Not yet? Then use the cal key for the teacher. Is this CLC C correct? People ask What do you want to replace x with? Right? So you look at me like this. Teacher There are a lot of these spaces to consider. yours right? I will consider that. the outermost space first. 1 to positive infinite teacher choose any value. You can choose any value you like. What number do you like? Can I have number 2? Ok. So let's see c x = 2. The result is equal to how much is this 12 right? positive or negative Positive. So What is your sign? Sign positive. See? Represent. Understand Are not? Representative. Then continue this 2/3 to 1 that is which number i choose now the you? Which number to choose now? Which number is in the middle? 2/3 and 1 This? Many of you will wonder if you know. is which number is correct? So teacher give what is the formula guys I want find the middle number right? Medium plus right? Add them up and divide them in half. 1 + 2/3 divided by halves regardless of the number any. understand? Add up and divide by half out in the middle? I take the average plus Now take 1 + 2/3 and divide it in half. teacher some? positive or negative Is this bowl positive or negative? Not yet. Ask positive or just sound? Positive. Exactly. Exactly. -1 to 2/3 choose number any? Don't know which number to choose then add them together split in half right? Let's divide -1 + 2/3 pair for teacher Understand? Add up divide that in half and get the average between. How much then? Yin or yang? Minus. Main body Continue negative up to -1 is some? Add it up for us we don't need. You need to know -2 right? -2 small more than -1 is fine right? -2 is negative or positive? Positive. Ok. done turn off computer Go. So then after we have like so then later all the articles other if you are having difficulty in To consider the sign of f', we just need to enter go to the computer and use the can key for the teacher. Do you understand? Teacher, please use this method. This can be done with any lesson. Are not must remember the left and right right and left are no different. Remember? And This way is always right, isn't it? this way To let the children realize one what is not every song interwoven where. Remember? This is positive this is still positive positive yet? But that is not positive. jump to negative You know why it is no sign change? Because guys, Number 1 appears twice, guys. x = 1. x = 1 yet? It appears twice. means it changed sign. Change the sign again one more time Understand? There. Position Thanh is still the correct sign, right? Positive to negative change sign once later change the sign again to positive. at it is a double solution. But as a double solution it will no sign change Do you understand? So x = 1 is the solution of this guy. Done 1 Is this guy's experience also? Listen it appears twice it is in normal form direction. Well, this is it for you. understanding is good Well then it won't. change sign The double root does not change sign. Ok. Yang goes up Yin goes down down. Head yet? Teacher told the students head yet? There. There. Copper function turn that side When e is 0 is fine too. Some zeros here. It still goes up the drain. often. Choose the answer teacher. Homomorphic on infinity -1 and 2/3 to positive infinity. This sentence sounds right but is it true or not wrong? Wrong. Wrong because there is a right match Are not? Do not use this. Are not use this Monotonous function What is -1 to 2/3? Going down is perverse. change? This false inversion. Function decrement -2 to 0. Where is -2? -2 This. -2 to 0 where? 0 here right? 0 in front 2/3 Are not? Does -2 to 0 have two signs? is wrong Not yet? -2 to 0 is both positive and negative Okay. Here it is. 3/2 is positive infinity. Correct Not yet? Why is this 3/2 positive infinity? guys? Is 3/2 greater than 1? 3/2 this range. 3/2 wire together. There then it must have a positive sign, right? From 3/2 lines both have positive sign? Copper gone? Already. Ok. Then sentence 10 is done. In y = fx has upper derivative r' equals -30x m 4 plus 60x² - 30? Ah Guys, people have calculated the derivative for you. you yet? We do not have to calculate derivative too. Don't be religious function again is wrong Children give 0 to teacher So f' equals 0. But if f' = 0 then I solve it's a little long Any subtlety? anything out? Huh? You guys see the subtle realization what not? Do you see -30 60 30? Teacher group -30 out inside out what else remember? x m 4 - 2x² how much is the square root? Wall plus 1? This hole? Then you guys see anything Special Are not? Right. This -30 equality times x squared minus 1 equals all squared. Have right guys? Oh, so guys -30 then this sound is just for you guys is bigger than equal 0. So what about multiplying? Minus Not yet? Right? But my y' is always negative. Not yet? What is the condition of the set? remember? Teacher, please give me the exercise book. What are the conditions? D no we are on R? Ah is always negative with If x belongs to R, what is the function? Always Inverse variable on R. Done? That means There are some special derivatives. We must be sensitive to that. yeah So it follows that f'x à fx is always increasing. always always reverse, remember to reverse above R. Ok done. I translate this here This error variable B is decreasing on R. That's if you know the derivative. The derivative is always positive or always negative, then the I confirm it is inversely monotonous always Okay? And it is derivative. complicated then you have to give equal to 0 and then create a variation table. there received tell me that Form 3 is for graph f(x). Now, form 3 gives the graph of f(x). Correct Not yet? If one is given the graph f(x) then How will you do? When the topic let the graph f(x) be the graph itself then You will judge the sign by being you. look at its shape Okay? Given the graph f(x), you don't need no need to calculate y' negative or j positive This is very easy. Very easy. Super easy Super easy. Super easy. Then what does my class need This? Look at the shape. Shape! It's so easy. This shape if go up go up this is copper variable. Going down is perverse. variable. Inverse variable. Is it ok? There, if people give You guys graph f(x) is super easy. You see the results right away. Like a variation table. Then look at the shape of the object. market f(x). Shape of the graph y = f(x). Miss Not yet? Among f(x) defined and continuous On R there is a graph as shown below. Is f(x) correct? Okay, okay. Then affirm Which of the following statements is false? Okay? Wrong ok? Then the function increases from 1 to positive you have to look at the up and down graph okay? There. Any This is the upward trend. Homomorphic this then goes down is this reversal. That goes up is copper variable. Okay then you guys look at this. Function monotonically increasing from 1 to positive infinity. From 1 This. 1 to positive infinity is monotonous. Exactly. Increasing function on negative infinity together to -1 right? Infinite Sound to -1 here go up exactly this. 1 to The wire is going up exactly. Then the sentence C is a decreasing function from -1 to 1. Let's see. -1 to 1 here. Look at this area right market goes down not down is opposite gone? Okay. My name is D. D is incorrect because the function is increasing on negative infinity. to 0 yet? That is negative infinity to zero. The graph goes up and down so it doesn't what could it be? Is dynamic. Okay Not yet? Ok guys fix this sentence. That's sentence 3. Which in the function y = fx determined on determined and continuous on infinite negative positive infinity and has graph Thinh draws the given function independently on the above. what time First, look at this sign. what is this? Let's just combine the dashes. From 0 to 2 of you look at this teacher. From 0 to 2 from 0 to 2 right? What does the graph look like? guys? Going down is the wrong inversion. This. Not monotonous. Copper must go up The sound v goes to 0. The sound v goes to 0. This. The sound v goes to 0. Okay? Oh city up yet? Standard This. Two strings together. Two wires here. The graph goes up. Not yet? Are the two wires the same? The graph goes up. Not yet? Still true. What is my answer? C. Is it ok? And from 0 is positive infinity. Here you look from 0 to infinity. This. The graph goes both up and down. wrong? ok Come on guys look Teacher, sentence 14 is as follows, you can see the sentence later. 14 huh? Given the function y = fx and continuously on the interval from negative v to below v same right? There is a function diagram. given the variation on which interval? Any look quickly for teacher what is this? Graph fx right? -1 to 1 Look here -1 to 1 this. The graph is up and down. down right? Homogeneity is only just go up This is wrong. -1 to 0. Negative from infinity to 0 here. Negative infinity to zero medium down and up wrong. 0 to 1. Let's look from 0 to 1. This. Here it looks from 0 to 1. Correct Are not? From 0 to 1 yet? Downward graph always wrong My score is D 2 to 5. Why? from 2 to 5 is like that, remember? Look at this. 2 here 5 it will be around this This. 5 here. Understand? Located here This. So this 2 year chart is going up. It's going up, guys. Type such a statue. ok Then from 2 to 5 it will always go up. That or the This one will vary from 1 to 10. infinite is certainly increasing from 2 to 5 huh? It is on an upward trend right? Are not? There. Now question 15. Given number y = fx defined and continuous on the interval r - 1 right? Get the destiny drawing program Which question is correct? This sentence is very scary typical guys This sentence, kids. How does the graph look like? Go down right? From left to right do you see going down That from left to right must go down So it is definitely naughty gone The paradox has occurred. But you guys oh, how does it change? It is different wrong about language Now, this symbol. What sign? This? Is the sign wrong? Is the sign wrong? guys? Is the minus sign wrong? Two notes and guys, A and D choose the answer any? Be sure to use the word and then. What if the teacher wants to change sentence A to If so, how should we fix it? You can not write, you can not write. is the defined set. Because this article How different is X, guys? Look at number 1 Does it cut the graph in half? X is different from 1. You see the graph of number 1. does it cut in half Make two parts? Split in half. So what should this post say? This? not recorded the specified set but recorded is each interval that determines each defined interval understand? You guys write that down as specific intervals. that's right. Ok then go to the final form together. Ok. Which form 4? Form 4. What type 4? Form 4 is for objects. market f'x. Remember that, kids. It's just different only one comma apart but it's different each other completely guys. Well if People give fx graph, you guys look at it into what? Shape right? Where is the graph of y = f'? be looked at in shape. Why? Religion That function only has positive and negative. Understand Are not? Not derivative goes up derivative go down right? Derivatives have only positive and negative only So you have to look what Are not view with axis Ox. Look at this ox. First above ox is ox must be y = 0 correct no kids? You have to look at the taste. mind with ox On ox then f' is positive. Where else below? ox under ox then f' remember? There is ox above that is yang below is negative. And he delivered with ox is f' = 0. Remember? You have to look at those that For example, here is an object. The function f' is as follows. FX here. There. Then look at this, at these are the intersection points this will be 0. Understand? Below will be negative, above is positive positive. There. There. Then you will see see, for example this is AB, BC then you guys Look, on what interval is it monotonous? For example from A to B. A to B guys see F' positive? F' is positive is copper variable. There, c goes to positive infinity F' is positive Are not? That is, because f' is positive, then homogenous right? You guys remember why because the derivative is related to the positive sign or sound What else does f(x) have to do with? up or down shape Okay Not yet? That f(x) is positive and is increasing. Not yet? Negative FX is inverse variable. I remember you as follows. Here is the graph of the function f' in the form of 4: The function y = f(x) has a derivative on the graph. f'. Read the question carefully. different sign comma affects the whole article. Graph like Here is the graph f'. Ok. Then any affirmation is an affirmation. correct? You look at me like this. Any when people give f' you have to compare with thank you teacher where is ox This is ox this is y how much Y = 0. Is this the ox axis? At what points does it intersect ox? Out of stock at how many points? The two points are point number and point number. how many here 0 and 2. That's right. You write let me say f'x = 0 is the intersection. Then x = 0 and x equals 2. Do you understand? There. F' intersects the line y = 0 at these two points Not yet? And look, below will bring negative sign, above sign positive. Not yet? So I will have a table. change This variation table. Friend Whoever studies well can be recognized immediately. varies from 0 to 2 at positive 0 to 2'. Okay Not yet? There. But if you want to be more specific, want to compare Compare the items and make a table for the teacher. X this, f'x and fx. Look at this, at 0 and 2 the direction is function equal to 0? There, 0 and 2 derivatives equal 0. Not yet? Then, guys, let's examine the signs as follows: Hey. Consider the following sign as you see below: This. From negative infinity to 0 the graph of f' is below? Below I imagine here is this land. This is positive Is this the underworld? Hades is below earth above right? This sound. That's it, don't be scared. Don't be afraid ghost This is positive and negative. Then the graph is like which one? FX is up and down? So you look here you can choose answer yet? So is that the answer yet? you? Look at this. F' is related to the sign right? f(x) is relevant down there So what is rock? 0 to 2. Many students said teacher, compare f(0) f1 how? Then here is this 0. this f(x) is f(0) this. Remember? 1 here. Right? F1 Downstream This. So you see F0 of F1 Who is bigger? F1 is higher right? Taller is bigger than. Really? F1 must be greater than F0 right? Board with children just compare these. Where else? Don't look at the board, just transform. Monovariant is just a variable. Ok. This sentence 17. Given the function y = fx with Is the function and graph f' as drawn correct? Which statement makes the correct statement? The look at this What does my graph f' look like? This? This is a special shape of f'. You see that the graph of f' is always positive. space? Always positive. never negative right? always what about positive Then the derivative of the number is always like any? Always monotonous right? there we are see is easy see. Is F' always positive? f'x is always big more than 0 is always on top. Should deduce the function how? Same variable on R on R Above R. Is the function increasing on R? That's because my f' is always on top. What is this discussion? R. copper coin Anyone who wants to compare this? What should I set up? Create a variable table out Okay Not yet? What to do if the inverse function is wrong? reversible? f' must always be positive inversely proportional to the difference. Okay? Already. If you want to check F1 and F2, you guys Make a separate table of variables for the teacher. Really then if the variation is the same then you can think always What is homogeneity? is x increase, y increase right? The bigger the better big. Number 2 is brother with number 1 here. Big 2 If F2 is greater than 1, then F2 must be greater than F1. Have understand? Same is same. 2 is bigger than 1 F2 must be larger than F1. Understand? There. Look I know that the function f(x) has a derivative. To be f' and the function f'(x) have the graph as shown beside. Okay? f'(x) has the graph as shown beside. Right. If there is a graph of f', then teacher I advise you to just let them look at the axis. Ox. Okay? here is the train Ox here Not yet? Hey ox here. There. Ox is y = 0. Now cut at how many points here guys? -2 and 1. Ok. f'x = 0 means x will be equal to -2 or x equals 1. So make a sign table for go Marking table. Variation table yeah x variation table This f'x and fx are like y' and y. Now, look at me like this. X negative infinite because determined on R to look is this right? My solution is -2 and 1 right? -2 and 1. This is positive zero. together. Do these two values ​​make f' equal to 0? this 0 Then you guys consider sign for teacher You guys look at the look on the Ox axis. You look at place x this then it cut in half? Is it from minus infinity to -2? is this always down? Then carry the negative sign. Remember? From -2 to 1, where is the graph? remember? Above is positive. Potential from 1 to positive v what about it? This is cut in half and it is still positive? That's positive. Meaning yin and yang. Because the graph It is completely above the ux axis. I will go Yin goes down and yang goes up like that this always So what is dan guys? yeah? what am i doing wrong yeah -2 to 1 the function is always increasing. Increase is the correct covariate Not yet? -2 to 1 always increases. Exactly. Go up is increase Correct Already. Decreases on the interval from -1 to 1. -1 at where? -1 where? -1 is greater than -2. -1 here yeah Right? Look at this. From -1 to 1 then what is the function like guys? Does it go up? Go up but wrong? Then B B is wrong. Jaw The positive number 1 is positive infinity. 1 extremely monotonous Main negative inversion from infinity to -2. Sound together -2 down the inversion yet? Okay. ok B is wrong. Then insert not gone Then thank you for understanding today's lesson. today? First find the domain. Corpse this decision This condition. First is the what are you? X is greater than -3, correct Are not? And 6 - x is greater than 0. Greater than equal to 0 correct Are not? That means 6 - x must be greater than 0. meaning x will be less than 6 right? What is the bottom line? This? -3 times mx memory equals 6. Is it ok? - 3 is less than x is less than 6. Ok Not yet? Yes. Next this. Domain Already. What will I come up with next? This? Calculate it. What is Y'? Y' you guys calculate What is this? y' will be equal to this square root of the x + 3 derivative. The What is the derivative? 1/2 square meter right? U' derivative root u derivative will equals u'/2 square root of u right? u'/2 square root of u right? The derivative is 1. 1/2 square root of x + 3 right? Then continue this custom What is the derivative of 6? U' 6 shots derivative is -1 right? -1 -1 on how much is this This? 2 square root of 6 chx right? U' on 2 rooms. There. Ok. Let equal 0 correct Are not? Let y' equal 0. Let y' = 0 always. What is the equivalent of this, guys? Now, change the clause to 1/2. The other side is 1/2, right? This square root x + 3 is equal to 1/2 square root 6/ x, correct. Are not? Prove that 1/2 reduced to its minimum is what to think This? Infer is x square root + 3 right? What is this? Equal to square root 6/x. Then what is the equivalent? This? What is this equivalent to? x + 3 squared up to 6 - x. So x equals how many here X will be equal to is 3/2. Okay? X = 3/2 yeah Ok. 3/2 yes Then x = 3/2 is satisfied condition? Satisfied. Satisfy the my condition is x is greater than 3 less than 6. Ok? Satisfied. Then we will have a variation table. what is this X This. Y' and y right? ok Already pay attention guys, I only take this lesson from how much From -3 to 6 right? -3 to 6 -3 6. Then you will remove all the parts. the rest go Just remove everything from -3 to 6. yeah ok Well, because x is from -3 to 6. X = 3/2 at between right? And this 3/2 will make y equal to what? Zero. Okay? 3/2 will make y' equal to 0. You understand? Are not? There you go. So now how do we judge the sign? Then this is the function y' this is a bit special one I will teach you how to use the computer later. yeah Click to review sign. ok Click on the check mark. In because this is not a true quadratic function in the left right? Any You guys quickly look at the instructions on how to check the marks. This. First you have to enter this guy enter. Enter this guy. Come on 1/2 square root of x + 3 - 1/2 square root of 6 x right? Firstly You have to enter the correct y' Are not? Now I want z to examine the sign y' right? Look at this. From 3 In parts 2 to 6, you take a value. any For example, you take the value of number 5. You can also choose other numbers. C this is the right button? This C x = 5. C x = 5 is positive or negative This? Negative number right? is it negative? Me It doesn't matter if the number is even or bad, what matters is The important thing is whether it is yin or yang. That is from -3 to 3/2 replace the number 0 correctly Are not? What is this? Positive. You understand Not yet huh? Understand how to do it? Replace a price any value, you choose any value, the If you replace y' with y, its sign is the sign of around there Right? There. Positive go up mic down Then at this point you can determine can i have the answer This knows answer yet? Come on -3 3/2. -3 to 3/2. -3 to 3/2 where? Oh, let me consider it. The sound of infinity comes 3/2. The infinity to 3/2 is not correct. Are not? The negative is not allowed. Because What does its domain do? Infinite Sound where? Also wrong because of negative sound the same positive infinity do not take, I give up Go. So is -3 to 6 true or false? -3 to 6 is wrong. -3/6 is both positive and negative. Wrong note -3 to 1. -3 to 1 right? Okay but. -3 1 here 1 it before 3/2 right Are not? -3 to 1. Ok. Here you go. 12 is D. First, people give advice. function above Then give f' then you have to give f' equals 0 yeah Is it ok? Then you pay attention This. For these 1 2 posts hey it's the high hat right Are not? Big hat right? Ok. This big hat let y' b= 0 what will x be? This equal to 0 for flat 0 then each equals 01 right? Because it multiplies together. X = 0 right? Then we follow this x = 1. What is X again? X = 2. What is X? X = -3 x what again? Equal 0. Is it ok? X = 1, x = 2, x = -3. X = 0. T for this is equal to 0 which is correct. Are not? So what do we get, guys? Then I will make a sky table. X this, y' and y yeah Ok yet? Now, what is x? This? This X must be from negative infinity. positive infinity Not ok yet? There. Already. Yin is infinitely positive is infinitely true Are not? What are the solutions? 1 2 - 30 - 30 - 3 this 0 1 2 from small to large yeah Ok Not yet? This is equal to 0. ok Already. Then you remember to pay attention to one thing. following Now you will use the computer. to check the sign to check the correct sign Are not? Use the machine calculate What is the computer like? Computer Hey guys look at this. I will enter enter to consider two to two with the same sign go But pay attention guys. Big hats here? The hat is very big. Hat then big hat computer it can not can calculate If that hat is big 2018, 2017 and the computer doesn't count Okay. So what do you do? Remember your hat! even Even hats are equal. 2. Odd for even 1. Even numbers equal 2. Odd numbers equal 1. yeah Remember? Now click on the calculator any. X - 1 right? All squared right? Even Even then equal to 2? Odd is 1 times with x tr- 2. Is odd considered an odd power? This is odd 2 odd numbers not 1. Core with what? Multiply by what? me? Even is considered 2, odd is considered 1. Is 2001 odd? even? Odd right? Then consider it 1 only. She wrote nothing. Multiply by what? Multiply x right? Even is considered 2 odd Consider 1. Ok? Then press this C button. This C replaces x with any value. period Let x equal any value. X how much is this? From 2 to below with the children what number guys Any number I scan number 3 go Picture 3. What sign is this? Sign positive. Understand? You will recite in your mind. head fast Understand? 2 positive and negative same is 72 positive. Continue this 1 to 2 squares of 1.5, you add them together and divide the square in half 1.5 yes Sound yet? Then continue this. Next C 0 to 1 then divide the dollar. 0.5 sound right? This next apartment -3 to 0 apartments -2 yeah Are you ready? Negative infinity to -3 cal next -4 Any size cage is fine. What is the sign? Negative sign yet? there is understand how to do it guys? There yeah Okay? See, do you understand how to do it? There, kids. just change any number you guys are not very just fast Between 1 and 2 units 1.5, 0 to 1 unit 0.5. Understand? Making Not yet? There, 14. Then even then equal 2, odd equals 1. Okay? Then and the Remember this, if it's an even number, it won't change. sign. Unchanged sign. The exchange sign. You don't believe it's you. Older brother guide you Even hat here. Even hats are the number 1 you see even hat? x - 1 2018 is even Are not? Does it change sign after 1? Are not change sign right? See? That's it Do these odd caps all change sign? Yin to yang yang to yin yin to positive? There you guys one remember one this property Even numbers do not change. odd sign, odd cap then change sign then we can do it fast understand? ok then I will have this negative then go down right Are not? The sun goes up, the heart goes down. go go up. Not ok what will we have here? X = -3 right? -3. E= -3 right? Then we What will it be? This place is f -3. f-3 This. Ok. This 0 is f0. Here here f1 This. You guys line down here f1. F2. ok See? F3 is this F3 0 F0 F1 and F2. ok Already. Then fate Which topic? wrong? Wrong. The function is increasing from -3 to 0. -3 to 0 is positive or negative? Correct. Right? This is it. F0 is greater than F2. F0 here. Is F0 higher than F2? Bigger is taller, kids. F0 large more than F2 F0 is greater than F2. Okay. yeah Then continue this. Inverse function from infinity to negative 1. Does infinity go to 1? Sound together to 1. The infinite sound here to 1 just down up down is not wrong. C wrong right? Continue this F3 F5 true or false? F3 like F5. F3 here Look at number 3 of number 5 here. F3 are the children the line down here is F3 This. F5 line down here F5 this. That's it. Is F3 like F5? Ready? F3 F5 is correct. Mean will be in order. The higher the bigger yeah ok So this is also true. Only sentence C is wrong. Extremum of a function number. What is this extreme, guys? we may have known each other since 10th grade with parabolic exercises. Okay? Any Teacher, this is theory. Lee this extreme theory If we assume we have been studying since grade 10 graph y = fx right? Then I will study in grade 10. parabola right? Right? Parabol then if the concave side is up and the concave side is rotated down like is this right? Then we have the concept. thought of extreme value. Extreme is a word to refer to maximum and minimum. Okay Not yet? See, extremum is short for extreme. great and small Maximum and minimum yeah ok Extrema is the abbreviated form of extrema. great and small Like monotonous is the abbreviation of monotonous and monotonous. Understand? Extremum is the generalization of extremes. great and small So when we look On the graph of f(x), you can easily see the extreme What is Dai? What is the maximum? Big is right Are not? Is it big? The maximum is convex point up. This is the extreme This. This is huge. Abbreviated as CD yeah This is huge. Ok. What minimum? To be concave point Is the point of peeing the baby right? Are not? Is the smallest point of the graph. There. The smallest of this graph. There. Thing this graph There. This is the maximum the highest point of this graph. Parabol right? Minimum is the smallest point of parabola There you go. Pole pee Pole grand. This is what we look like. The Will you see the maximum and minimum? So if the teacher gives the graph of the function f(x) as position Hey, the whole class, give me some. minimal? Now look at the picture. See Which point is convex? Graph y = f(x) yeah Come on pot up is a maximum this. Pot is it maximum yet? Concave is the extreme pee? Concave you understand like that valley There, concave is extreme. pee This minimum. This minimum. This this extreme. Maximum. ok So it will how many maxima How many minima? Two poles what is the pole and pole? Two minima. Okay? So how many extreme values ​​are there in total? This? Four extremes. 1 2 3 four correct extremes Are not? The four extremes are divided into two. big and two small right? There. Extreme is the common of maximum and minimum. This four poles four poles Four poles treat Okay yet? That's what it's called. So You can have the most intuitive view that is looking at the convex fx graph is maximum, concave is minimum. Okay Not yet? There you go now let me understand deeply than the class I wrote for the teacher like after What? If x0 is an extreme point, then it is an extreme point. There, remember extremum is the general term for extremum. great and small I abbreviate it as extreme treat If x0 is an extreme point, the point extremum must satisfy two factors after. This must satisfy the following factors: after. What is the first? is y' will change sign change sign change sign through x0. y' sign through x0 and y equals y = fx0 exists at at x0 there For ease, y = fx0 exists x0 yeah that is, substitute x0 into the function fx of I am satisfied, understand? And he did. what? Change sign. You remember the teacher from the beginning mark Can I change the sign? Change sign. Change sign. Any. Apart from 1.1, look at me. Drawing teacher one here Draw the maximum like this. X This. This FX. F' this. FX hey This is the negative infinity for example. positive v and x0 This. Look at this. Case number two best If it changes sign, you see. see y' change sign right? It changes sign from positive to negative From positive to negative. From positive to negative, the graph is positive. how are you guys? Yesterday I learned What does the graph look like if it is monotonic? Go up what about sound Downward. This is it fx0 f 0. Okay? So what is this guys? yeah? What pole? What extreme? If you change the accent mark What is the pole from positive to negative? Maximum. Right. There. Then continue the case this monday X hey, f' hey, f(x) Now, this x is negative. same Here x0 is positive infinity for example. yeah There, if you change the sign from negative to positive then you know what it is? You know Already. What about sound? go down right Are not? Yang went up. ok X0 then here is f(x)? f(x0). So what do you guys have? If change sign from positive to negative what pole is this? Maximum. If the sign changes from negative to positive then minimal Okay? There you go. Remember here If the sign changes from positive to negative then it will be maximum from negative to positive. In because going down and up, I see the picture Drawing is what I see most intuitively. Okay yet? So the teacher is as follows. If f'x f'x is this right? this f' change Can you change the sign? From positive to negative. What is from positive to negative? This is huge Pole grand. Ok. There. If f' changes sign from negative to positive positive is the pole pee Remember? What is positive to negative? Yang goes up and yin goes down. There, right. is it maximum? What minimum? Change the word mark negative to positive Yin is down, yang is go up. Is this minimal? Maximum Are not? There, you have imagined it. this yet? Change the sign from positive to negative is maximum From negative to positive, this is the extreme. pee Okay? There, the minimum is this value guys This is minimal. This. This is huge. This is huge. And here is the pole. pee Remember? What about cooking X? What do we call this guys? X is the point extreme What do you guys call this X0? Any Don't be confused. X0 is called a point. Okay? The maximum is just fx0. Okay Not yet? The minimum is also just fx0. Okay Not yet? You know how extreme it is. You know what the extreme is called? is off what? Write long is the price maximum value Maximum is the maximum value. People just write it as maximum. Pole What is urine? Is the minimum value. And people just write it as minimum. Okay Not yet? There and you miss the teacher. That's extreme. Most people only talk about Y. That said about Y yet? What about X? X then people call X is the extreme point extreme point extreme point pee Okay? Where is I? Not yet? This is what people call a point. people call it extreme value Just the medicine You have to distinguish for me. Because when people ask people can ask the What is the maximum point of me? how much is the minimum ok There. Still y is called the maximum value. Price maximum value or abbreviated as maximum. Pole great Okay? or extreme value pee Pole sub-two with t being the minimum. Okay? There. Where else? If a point M is X0 and Y0 ok? X0 and Y0. X0 and Y0. Remember? Here is a point M X0 Y0. Wallet For example here is a point M. Point M here. X0 Y0 It includes both X and Y. Then you know what do people call it? This is just just terminology To distinguish, we ask stupid birth control questions Come on call To be extreme point extreme treat of the function graph. Function graph number. Remember this carefully for me from the function graph. number People ask what is the extreme point of the graph? For functions, you have to say the extreme values ​​only. Is x and y ok? There. Where else? maximum point of the function k of the function teacher just say function The function if is only Is the function just x ok? That's it jaw number. Ok. The function is just x or y but from graph You must give the teacher market What do you need to have a graph? There is no graph without x and y. Right? If you want to draw a picture, you have to be ugly. There, This sentence is very well researched. Teacher keyword 1 There. Already. And the whole class Did I notice anything? Look at this. Look at me, kids. this table Minus 1 you know This is the maximum point that makes y' opposite in sign. right? Well, the minimum is the same. let y' change sign from negative to positive. This is positive to negative So for you guys at What sign will my f' be? It will is number What? 0. Well 0 is correct but only half correct Stop. At the pole at point f' it can be is either two bars or a zero. One of these two. Okay? That's one of them. two dashes and 0 is fine too ok At this point f' can be 0 or two bricks. Ok. So my class Write down for the teacher a property as follows. At the extreme point. at the extreme point. In extreme point product0 can be 0 or 0 is determined Not yet? Can be equal to 0 or not defined as a colon. Okay? In At extreme points, the derivative can be 0. or what? Not determined. Remember? The symbol can be a double-diagonal letter or What are the letter symbols? Number 0. Children remember that teacher The first posts are articles with a lot of theory and The teacher must write clearly for the students to understand. I am not mistaken. Remember? Then write it down. go The same thing is certain. theorem like after. Let's go through this theorem to find out the end of the theorem. theory This one is all right. theory So the theorem is as follows. To be If f'x fx ok? f0 is equal to 0 and f' is like the second derivative and if x0 is positive then x0 is extreme points for you guys extreme What if f' = 0 and f' is greater than 0 then for you is the extreme point what is the maximum or extreme mini maximum or mini minimum now Guys look at this f' 7th place is the derivative of the first degree two right? The second derivative will be represents the trend of the graph. Coin direction of graph Trend trend of thing market Okay? The second derivative is the direction of graph Okay? Then you guys oh, f' is positive f' is positive second derivative meaning uptrend or downtrend reduce the children remember? Yes, positive is increase right Are not? So when does it increase? So then according to you is maximum or minimum then tend to increase? Maximum or minimum of the uptrend remember? Of course it's small, it's small, it's the ceiling, baby is the floor then i have to increase right? So then there is a tendency to increase which is the pole remember? Minimal. And the giant has coins which direction? Discount right? There, f' it for the trend you remember so as not to make mistakes mixed Okay? Already. F' increases, f' is positive means uptrend. Increase must be minimal It just grew bigger. Not yet? The baby has just grown up. Correct Are not? then extreme this continuation If f'x0 is Not yet? If f' x0 is less than 0 then What is x0 point guys? To be extreme point great Okay? If f' = 0 and f' is small greater than 0 then it will be a maximum point. In Star? f' less than 0 is a correct downtrend no kids? Less than 0 is the trend reduce. What is the downtrend? When it is maximum then it will tend to decrease. Okay Not yet? People are extremely prosperous then decline right? Do you know that? When it reaches its peak it will start to head fall down right? Where is it when it hit rock bottom, what the hell is going on? you guys are the same as the life human life When reaching the top high it will go down and when it reaches the bottom I can't take it anymore I have to do life over again then it will be my life will be happy again There done when i get taller then I was a little careless and fell down again. right? That's life maximum minimum Okay? That's it! my life is something it will never be the same flat right? When I feel like studying very good but suddenly a bit off today I'm a little lazy and I'm not good at studying. drop a little There, go down a bit and you will see. this is not good, i have to study well more and more increased. Just like that right? Fluctuates like that. Lifetime is always a maxima-minus graph. So what should we do Try to achieve those that extreme, the new peaks must be higher than the old peak. Remember Not yet? Well, I may be worse but next time try harder old times Okay? Well, when that happens then I will fall down then it just falls down come down here Right? Down has where are you guys just keep going straight like this This. There are people who live it not bumpy like this Life is so rosy straight like this, straight like this. And one day come to this place meet with trouble try to fall one time can not find life again always. Right? When I try to try try then for example go here teacher got me Suppose something happens to me and I fall. I just got down here. Can't fall can i add more? That's it. I can start over. I can start over in another direction. Right guys? There was a time when I kept going straight when one thing fell off, it fell off never to the end That should be our life sometimes has difficulties difficult, there are challenges then do for myself more and more more mature Remember? there thing my market, my life is like this. Okay? Come on, teacher, finish the theorem. this last any. Ok. Multifunction awake. Multifunction awake. Polynomial functions are cubic and quartic functions. 2x level polar treatment x0 then f'x0 equals 0. You guys remember the teacher. If the polynomial has a radical at x0, the derivative will be equal 0 huh? If the polynomial function has an extreme value at x0 then f' = 0. Okay? Polynomial function has value at x0 then f' = 0. Okay? Already there. These are the theoretical numbers we let's copy it Now what teacher? Now, part two. Post which episode Part two of the article which episode The teacher will divide the exercises into different types. give them the types they often see. Okay? Form one is easy best. Form one is given a variable table heaven Given variation table. ok Given variation table. Ok. And if the problem gives a table of diseases If it's natural, you just need to remember it. What is a radical teacher? Remember the extreme teacher is What? First, f'x0 changes mark through x0. Ok. And the second one is fx0 exists in. Remember these two conditions. Okay? f'x0 changes sign. Okay? f' change sign through x0 remember teacher from change sign and fx0 must exist, not satisfied Only when these two conditions are satisfied is it the extreme point. Remember, x0 is the pole point. Is the treatment ok? You have to satisfy that condition. That is a hint that the teacher gives the table of variables. the angels and the children look at the teacher like this This gives the function y = fx with a variation table. as follows. by an affirmative Which of the following is correct? I analyze. Which function reaches a maximum at x = 2? where is the pot up is extremely large is not extremely large The standard here is x = 2, which is not correct. ok done always teacher continue to analyze the answer below The function is maximal until x = 3, wrong x = 3 where x = 3 is y maximum? is y maximum not Why must it be calculated? x The function reaches a maximum at x = 4 which is incorrect. What is 4 tense? True minimum Are not? The function reaches its maximum at x = -2 which is incorrect. Already. Peace is a. Ok. 1a. let's see example 2 Given the function y = fx with a table of variables the picture and you have to choose the correct one affirmative Okay? Come on guys look here First, you analyze give me these points X0 x1 and x2 see which is maximum and which is extreme pee You look at point x0 for me. Does Y' change sign? Is there a sign change? Does point Y exist? Does exist. Ignorance is existence. Don't know what is existence Ok. This is the minimum. Ok. This one is minimal. Does X1 change sign? Has the sign changed from positive to negative? Changed sign. Ok. So this is minimum and maximum. Okay Not yet? X0 X1 here. Yang to Yin up high is the bullet Remember me at What is the extreme point? y' can be 0 or what? Indeterminate is polar treat Not this one what kind Where else do you look at the score? x2 for this teacher. It satisfies the sign change Not yet? But does it satisfy the second factor? what is two What is the second factor, guys? yeah? Does f(x)0 not exist? Look, this place exists. Are not? Does not exist. Does not exist. In because two dashes do not exist. There it is. only allowed not to exist in y'. And yet it must exist. Do you remember? Already. So what is my sentence? One pole great, a small. One extreme, one extreme pee A then have 2a. Remember? That guy this is nothing okay? Want to be maximum or minimum must change The sign and point I must be determined. remember? There cannot be two lines in the wye. Then for example 3 for function y = fx with variation table picture. Ask how many points the function has extreme? Let's analyze it. Bag how many extreme points? Extremes are general of maxima and minima right? Count This is at point -1. Ok. Y' standard sign change Not yet? Does he exist? Okay. Guy This is calculated this. This one at Has this zero point changed sign? But he does exist. why? Not exist, not allowed calculate Remember? Continue this. At point 1 Has this changed sign yet? Exist yet? Ok. So it will have two extreme points. Remember? Two maximum point is equivalent to two points extreme value B One extreme point, two maximum point Okay? The maximum is common of extremes right? The extreme is maximum plus minimum. There it is. there will be two maxima. Ok. Then this will be is bipolar grand. Two extreme points. Ok. Now, look at example 4 for me. In y = fx, you practice determining the negative. infinity to 4. Sound infinity to 4 here. Already has a second variable table plotting the extreme points of the function. You give me some good marks treat here? This must be extreme. Are not? Change the sign yet? Change sign y determine this standard 2. Change the sign yet? Y determined yet? Standard. Don't care about that yes Remember to change the sign. Mandarin weight change At this point 3. Change sign yet? Determined? Already. Ok. So dad The score of this father's point is calculated Are not? Do not have. What should it have? Pole maxima it must have two branches. Miss Not yet? This one has only one branch. Shouldn't have waved it up here a little more. then this is considered extremely extremely pee Okay? There it is not not meant maximum minimum must be in the middle? In the middle of this in the middle of this ok? Where are these two heads? yes Okay, okay form. two. In this second form, you have to give the function y = fx or f'x. Now, let y = fx and f'x, then our class must do know what? Oh guys you have to What? Must make correct variation table Are not? Make your own variation table like the lesson yesterday No need to create your own variable table thank you teacher Self-made self-made table of variables heaven Remember? Create your own variable table heaven Do you remember how to make a table? calm? So step one is What? What episode are you looking for? Identifiable set Not yet? The second step is What? Is the calculation correct? Calculate this for y' = 0. Right? There. Let y' equal 0. There. What is the third step? There. You find points that make y' = 0 or y' not identify. Not sure yet? To I filled in the cross-section table. me. This X, y' and y are entered into this table. Okay Not yet? There, you will line up in order. gradually increase Has it increased yet? There, you have to put it in order. This x is in ascending order, not arranged haphazardly Then I finished just discuss Step 4 Conclusion Stop. Conclusion only. Look here are the I will conclude. Remember? There, teacher. will model a little essay for you guys to do You can understand the theory. The first lessons are the bricks. so i do it really carefully lesson for teacher Second form for function fx and f'x We must be independent to be right. Are not? I just saw it myself. But for this function you can not see no Ok. Now, step 1 is the identification set. what is it? D = R. Polynomial function for which D = R. Step two calculate y' for teacher equal to what? 3x What is the square of the derivative minus? 6x equals 0 for teacher X what any? How many kids? X = 0 and x equals 2. Not yet. Let's make a variation table. X this, y' and yes Right? Now, at points 0 and 2, arrange this teacher 0 and 2. from negative to positive. Why? What function is this? Quadratic right? It has only these two solutions. And practice sinh is R. These two points make y' equal to what Zero right? Let's consider what sign Quadratic function then I How to check the sign? In the end. Exactly. 3x² squared x is positive. So in is negative, outside positive. So how? How is Duong? right? Go up, go down and go go up. Yes, right. Not yet? Same as last post. Come here is the same as yesterday's post. But you guys Hey, today's lesson is you will add a a little for you as follows You guys change the sound Come here for me and I will help you. Conclude what is the result? Let's calculate the limit. Change sound how many in here What is the power of infinity? By negative sound together. understand? Substitute 0 for y how much are you How much does 0 get here? Instead go find this one. Replace 0 with how much is this 2. Replace 2 here some? 8 c 2 is 10 minus 12 is -2. Ok Not yet? Replace infinity here, teacher. how much Is the power of 3 positive infinity? Just memorize it but. ok See? Come here of my class will conclude as follows. Attention this conclusion. Jaw what is this function, what does this function achieve? This? Maximum. You have to conclude for teacher The function reaches a maximum at x = 0. True x = 0. Has the function reached its maximum? Jaw peak number great x = 0 right? And what will the maximum y be? you guys? What is the maximum value of this medicine? Equal 2. Remember? Already. The function reaches a pole. x equals some? At what x does the function reach a minimum? Is this minimal? What X? X = 2. There. X = 2. And y is minimal. This is called the y pole. little y minimum will be equal to -2. ok That minimum is at 2. Y pole The maximum -2 is 0. x = 0 at y pole equals 2. What is the answer, guys? yeah? The function reaches its maximum from x = 2 and x is equal to min x = 0 max wrong Already. Minimum at x = 2 maximum x = 0. Ready? Already. Minor is 2 at maximum is x = 0. Here is B. BB right? There. Actually, you can see it here. read the correct answer but the teacher guides you do you remember how to present it? There. let's see Example 6. Given the function y = x to the power of 4 - 2x² + 3. Which statement is true? Remember the children step What is a defined set step? D = r. The second step is to give the teacher a thumbs up. any. What is this guy? 4x hats 3 right? minus 4x equals 0. So the I type on the computer for the teacher and the group on death row always 4x this x average tr- 1 equals 0. So what is x? x = 0 or x equals plus -1 right? Okay guys make a table of nature X this y' and y then This infinite sound you fill in this value -1 smallest 0 1 positive infinity 0 right? Then you examine the sign like that. which one? Remember how to check the sign Are not? How do you judge the signs? Outside same right? Is the coefficient 4 positive or not? Are not? Positive. So you replace the positive with negative. same In general, you guys change here. This. Or you look at the coefficient of y' that's OK. Let's hide y' correctly Are not? You guys replace the positive and negative in here. What is that? positive or negative Yang and I are in. is this positive So it will be positive. Remember? That's the mark. That function This is a comfortable knitting pattern. Interpolation function level 3 level 4 knit this mark Already What is positive? The sound is going down. Yang goes up yin goes down yang goes up. Ok. So how many extreme points does the function have? you? These points are 1 2 3 right? there 1 2 3. The extreme point is x 1 2 3. The three points Are these all extreme? There you have it. can look at the concave or convex point look at y' change sign also. Answer what am i A sentence 6a. A this sentence People do not ask exactly how much so much then our class does not need to change you can come in Just circle it. This 2m4 right? Meaning is it the This function I am wondering now is I don't know how to mark it correctly. Are not? Then I will change to positive. hey hey I will replace this one This. Okay? I replace the positive and negative together. is this positive Are not. Same in here must be positive infinity 3 is positive? As this will be positive Not yet? Then I will knit the mark. Knitting yeah The cubic function with three roots is Knitting is comfortable. Number of solutions by the number of steps then the mark is woven. Number The solution with the number of degrees is signed. Three Here are three cubic roots with alternating signs. Not yet? There. Number of roots Number of roots that are equal to The number of steps is indicated by the symbol. Number of steps then you knit sign. Comfortable stitching. So the equation If the third degree has three roots, then it is certain that the sign is interleaved. ok Quadratic degree has two roots. got it? There, comfortable knitting. Number of solutions is equal to The degree of y' is definitely interlaced. yeah If you want to enter my computer instead special values Understand? Miss Are not? Yesterday's post? Enter y' as computer? Then can the values especially here. C replace and 2 here. Replace 0 + 1 and divide by 2. Remember three days before? I change special value It's okay to enter but it's a waste to replace. time. This lesson is easier than the punctuation mark. right? This example 7. The function y = x to the 4th power - 4x m 3 minus 5 is correct. Are not? Take point x = 3 as the minimum point. How to do this? First step 1 You just make a sign for the teacher done. Now d = r. Y' equals how many? you? What is Y'? This? What is 4x? 3 - 12x² right? Equal to 0. What group do they have in common? This? What in common? What's left in the vase? What does 4x² mean? X minus what? 0. Yes right? And this song guys equivalent x equals some? X = 0 or what is x? 3. Is it true? like this? Then you make a table for me. This article is definitely not punctuated. Do you know why? What level? How mystical? Level 3. Some solutions This? Two solutions. Degree 3 with two roots prove what? A duplicate solution. understand? Degree 3 must have three roots right? Then surely a solution The duplicate is this x vase. Have is it x squared? It is reset here. This. X squared is x like x. That's level 3. If there are two solutions, let them be the same. so you can not knit the mark. then no For the cross stitch we just need to choose one. any value Yesterday the teacher taught the students I pressed the computer already. Those are easy lessons. then the teacher does not want to use the calculator because It takes longer to enter the computer. y' and this medicine This sound is infinite. Extremely positive This. Ok. 0 3 0. Come on teacher, please guide me. My class checks the signs as follows. Which post? can do it all Come on guys look at him for this teacher Yes this is correct Are not? You enter this into the computer. Go. Enter this into your computer, guys. Just replace the special values. For example, if the teacher replaces the number 4 here, it will be considered. sign y'. Teacher, replace 4 here. Do you think in your head positive or negative? The you replace 4 with y' Because I want to consider y' sign. You guys replace 4 here with positive or sound? It's easy to replace the teacher. Replace 4 come here 4 x 4 to the power of 3 - 12 by 4 squares should i press positive or negative Positive or negative positive right? Come on from 0 to 3 guys Enter a value in the middle of the product? Number 2 huh? Replace number 2 here for me to see. positive or negative Click this yourself. Children can also be replaced. Replace 2 here. Replace x = 2 come here. Sound yet? Right? Continue negative to 0 then teacher can replace -1 Are not? -1 ok? Any number. Understand no? You guys replace -1 here. how much teacher Get your computer and click. Use enter the key This CLC. In CLC This. Yesterday the teacher showed how to press Already. Yin or yang? Minus. Oh you see Not yet? It does not change sign. Not yet? Double solution. This square later you will understand this is a solution double Double roots will not change sign. Because it is square and always positive Are not? There, remember the teacher is a double solution then no sign change This dual solution is either Even multiples do not change sign. Will not change sign, unchanged sign and zero are extreme values. No sign change then it will not be extreme. Not the extreme treat Okay? There, the double solution is Even multiples will not change sign. is not extreme. For example, what experience guys? x square is not this extreme X - 1 to the power of 4 is not either. is an extreme value because the fourth power is always positive, right? understand? Then you remember the teacher. yeah Here are the double and even multiple roots. This You see x squared is an even multiple solution. even? Even hat? Why? do you know Extreme, remember! what is teacher is y' change The 2nd caret is always positive, how can I change the sign? Okay? understand? That's always positive. It cannot be changed so it will not must be extreme It only has odd multiples. is just extreme Is that it? Yin goes down, yang goes up. ok The sound goes down like this. There continuously So what is dan? you? The function has x = 3 as its minimum point. X3 minimum. Ready? This is minimal. This. Exactly. What is the answer? A 7A yeah Remember? double root and root Even multiple here it will not change sign right? So if you want to be sure then what remember? Substitute each value into the string. This. Okay? Other articles are similar. self Then wait. Step one is me doing it what Find the right conditions, right? What are the conditions guys? The condition is x different? X is different from -2 right? Next Steps calculate y for teacher Remember the formula fast? Now cross product down minus cross product up. 2 x 2 is 4 minus minus becomes plus right? So 5 is 4 minus -1 is plus 1 is 5. 5 what is this? What about X + 2? Peace of mind is the direction always positive? Right? The Do you see your y' always positive? Always positive? But I still Draw a table of durability to understand this clearly. Board of argument this day x this, y' and y this. Okay? Is it? is the condition x different from -2? -2 from infinite negative to infinite positive. Any, -2 in the sample right? What are the two samples? guys? Two bricks. So you will give teacher two dash here Right? Two bricks but. Y' is also in the denominator. Y is also in the sample yet any? What is its Y, guys? Always positive. There. What about Duong? Go go up. Going up right? Duong then go go up. So you guys look at this. extreme Are not? Right. There is no extreme value. Pole What is treatment? y' changes sign. Yes you see always positive then there is definitely no pole cured right guys? That's what I drew. Just for fun. Ok. Are not have extreme value. Okay? Y' is always positive and constant. sign. No sign change, no value. Okay? Already. What is my answer? This? Baby. Question 9. Whoever chooses C is wrong. always. The answer is A. Why is that? So? You know? The maximum is X or Y. your maximum is X or Y. Read it again. Re-read the maximum theory is x or Y. Theory is X or Y is y. Right, maximum is short for maximum value is y That means you have to find the maximum y. for teacher Remember? There, from the new point what is it? The new maximum point is x. Remember? The new minimum point is x. There, This sentence is very easy to get wrong. easy to make stupid mistakes Not yet? Okay, step 1, d = r. Step 2 what is two y' Calculate the teacher equal some? This square x minus 4x plus 3 b= 0. So what is the equivalent of x? 1 or x = 3. Make a variation table yeah X This y' right? Infinity this 1 this 3 positive enter together. Right? What degree function is y'? Quadratic is only inside left and outside left. yeah Quadratic in left and right. Yin, inside left is yin, outside is positive. Yang goes up, yin goes down down, positive up. Okay? Already. Minus What is infinity? Sounds great right? The sun is extremely What? Is the sun infinity yet? Already. Ok. So you fill it in for me. People ask is the maximum is to find y Not yet? Not looking for x. Children no x = 1 is wrong One ask strong is y This is the maximum. Okay? Well, how many do you get by replacing 1? How many do I get if I replace 1 here? 1/3 yet? Already. Already. So what is the result? 1/3 a yeah And the person who asked the minimum question what is slime? Replace 3 here for me how many guys? How many teachers can be replaced with 3? You change. Replace number 3 here for me equal some? How many? Replace teacher with bag how much? Change it yourself. What sound? -1. Oh. Already ok There. Where else do people ask for the minimum? is -1 Minimal. Minimal. Pole The minimum y is -1. Okay? There. There. People ask maximum or minimum Can people talk about him? Miss yeah Don't be stupid. Then give go Question 10 class I look for teacher. This sentence is quite a trick. Extreme point subgraph of the function graph. Repeat the reason What does theory mean guys? Graph What should the function be? Yeah, it has to be a point. XY. Remember? So, A is wrong, B is wrong, C or D only. Remember? The graph is right there is X and there is Y. Remember? There, there. Calculate go teacher Calculate y' any? Oh, let's practice identifying. What is D? R minus what? Minus 1 right? Because X different 1 Right? D = R - 1 Already or you can rewrite x differently than 1 that's OK. Come on, how much do you pay for the teacher? how much This? What is the derivative U/V? Come on, U/V derivative Repeat hey, what for? U'v minus V' u on V square? Ok. Now, calculate the derivative. always give teacher U' is 2x minus 2 times x - 1 minus this teacher has equaled some? By 1. x squared minus 2x c correct? V' person u split sample not right? So y' for future how much is it guys? Shorten teacher let's go Let's shorten it. V sample vase Go. X - 1 to square. Let's shorten it, teacher. any. mumble 2x times x is 2x² squared minus x squared is x square - 2x - 2x is -4x plus 2x is minus 2x correct Are not? Can you count? Or you guys let's release it. Who can count? good yet Mentally multiply coefficients What is -2 times -2? -2 times -1 is positive 2 minus 5 is minus 3. Ok Not yet? There, split the sample. jar. Who can not count, teacher please write Let it go, man. let go There, then equal to 0. Still for y' equals 0 right? What do we get? Click computer for teacher X equals some? What is X? 3 and what? 3 and -1. Ok. Then make a table of heaven you guys yeah. Average table. X this y' and y. Okay? Look closely at the board. like teacher after. What is the condition x different from? Other 1. So from infinite negative to infinite positive 1. So one is the one who gets two strikes. for teacher like This. Ok yet? -1 all 3 fill in the blanks are those value that makes y' equal to 0 right? -1 and 3 here makes y' = 0. Then consider the sign as any? How do you check the signs? Let's look at the sign. guys What degree function is y'? The sample is square right? Square it up throw away right? Always positive don't care about the mark anymore what is it What degree function is this? Step no? Children is this a quadratic function? Degree function two tenses in left and right The number 0 will be left. This is x squared. this positive In these two zeros. Remember? What is the opposite sign? In two zeros What is that? This is negative. Remember? Outermost sign. In the outer left. Here you go guys see the two zeros? In opposite sign, outermost sign But now my class enter into computer try try try also waste of time right? We then try to make the exercises harder, they are difficult sure, it's complicated. These articles wrong This is very much. What about Duong, guys? Go up negative go down negative go down positive go go up. OH. So guys, the minimum point is what point Minimum point right? Is this x = 3? This is minimal. So How much do you find for the doctor? The I will replace you. 3 y equals what This? Teacher 3 here can replace 3 with what? How much is 3 here? Change it. Replace 3 come here guys yeah? Replace this with 4 right? On one's own change Children can change. This 4 this Ok. So I will have the minimum point. of the graph of the function are 3 and 4. Not yet? There. So m is 3 4 is the minimum point. Extreme point remember of the function graph. Okay? There, the graph of the function must be is a fish. Yes. Remember me, teacher. Tips This is not a trick but a formula. U? The reciprocal of the derivative u will be u' times u divide by the opposite value of u. This is my class remember teacher add a formula to calculate value absolutely new yet? The derivative will be This is equal to u times u divided by its opposite value. Not yet? U can be cured yeah Here u like u' divided by minus opposite u is correct Are not? So is this post done? Finished Already. What is identification practice, kids? Does the article have any conditions? Do not have. D = R. Is it ok? Continue this. What is Y'? guys? Answer for teacher any. U' multiplied by U divided by the value of U. Get Not yet? Hey u' this is U. You see This is U. Come on, let's calculate the derivative. U' is some? 2x times what? X squared - 1 divided some? Divide by divide by correct reciprocal Are not? What is this absolute? X average - 1. Ok. Let it be 0. What do we get from 0? This? What is 0? X equals some? X = 0 or x equals x = plus -1 Not yet? ok U' times U divided by U. So to create a variation table for the teacher Not available. This table turns heaven this, y' and y this. Infinite Sound This. -1 this, 0 this, 1 positive infinity. Any This sentence is very easy to get wrong. One more attention. The example posts are all plugins. the easy things that you can easily get wrong yeah But the example in class is that you guys. pay attention like that. Which one do you choose? Children look at this Many of you filled in the number 0 correctly wrong? Now y' = 0, teacher gives these three solutions right? But guys, this is the model. must be different from 0? That is x b - 1 Hey, this negative plus -1 solution, guys! This experience replaces it with what it does. how are you guys? make it uncertain right? Teacher still make the sample equal to 0 but if the sample is 0 then is it determined? Unknown is two brick for teacher in y' but only two bricks in stop it Why is its domain is d = r substituted for y still satisfied right guys Replace y is still ok Not yet? But instead of y' it's under the sample Are not? So is it undefined? Already. Ok. What about point 0? No change If you are satisfied, please fill in here. just 0 Right? At point 0 then replace y' still satisfied but still equal to 0 Not yet? Because the conditional pattern is x different just add -1 Okay? Here you go guys look at this Let this be 0 then it is double Can this guy get along? So this guy must be incorrect. determined Okay? There. Then guys. You Now I just check the sign as usual. guys You guys check the mark for me. You enter this into the computer for which teacher Is the treatment always positive? Older sister always positive You guys give up. Treat always positive. You enter this expression into the teacher's hand. into the computer. Now type your hand into the machine. calculate to see which can be which value positive Enter to calculate the value who import this guy please help give me any value you try Tell me if the number 2 is positive or negative. I see 2. here positive or negative yes 2 here positive the you replace for me 0 to 1 guys try 0.5 replace 0.5 here positive or negative you guys try Change it yourself. very easy Stop. This replaces the number of jobs. just number Negative -1 to 0 is positive. Female is negative. Okay? There, it will go down. You guys do it yourself. Select special values ​​to replace yeah There. Ok. Down go up. Ok. There, remember that. It still is determine This is the minimum. This is extreme this big This is the minimum. Okay Not yet? Y' is not definite but y is still identify. Okay? Y still determined instead It's ok to come here. Okay? There. This post very noticeable Remember this formula, teacher. Derivative treatment bear opposite to. Then what are you guys? What you guys? The function has three extreme points. 1 2 3 three point value. What is extremum? To be point that makes y' opposite sign. Change the sign yet? And determine at y0. Determined at y. Okay? Ok. Okay Not yet? Go. There. This sentence is very good. Any you guys yeah. What is this function? Is this a polynomial function? Example sentences 12 this. Knowing the graph of the function y = x m 3 - 2x² + ax + b has extreme points at a1 3. Okay? Already. Then the value of 4a - how much is b Then you remember the teacher our final theory notes. The polynomial function has a maximum value at point x0. This is x0 right guys? This is x0 Are not? Is this x0? There. Polynomial function has extreme point electrode at x0 then f'x0 how much equals 0. That's right. Exactly. The polynomial function has the derivative at this point. will be 0. Remember? We will have this we will with this f'1 will be 0. Remember? There you go I remember this first. Because of the extreme point Here point A13 with note is x0 here. Okay? This X0 1 is going to make the Tao Is the function equal to 0? Extreme is extreme point that makes the derivative equal to zero or zero identify. But definitely determined then this is a polynomial function. You have recognize? Then definitely score What will the derivative be? 0. Okay. Not yet? Then calculate the derivative for me. Yes some? How many you? 3x square minus how much? 4x + a is correct Are not? Is it equal to 0? what How much can I get in parentheses? Any any? How many do I get if I replace 1 here? This is 3 minus 4 th 3 - 4 + a = 0. So what is a? you? A some? A equals 1. Not yet. On the other hand, guys. me again what else is the nature? Hello guys. Thing The graph of the function has this extreme point. right? Suppose the teacher draws a graph randomly. function like this If it is extreme is like this, point A Now I ask you, does point A belong to? zero graph? The extreme point of lying on the graph of the function is not there. So A belongs to f(x) belongs to y = f(x). So what will we have? This? Replace point A with f(x). here Replace this Not yet? Replace point A 13 here. then you replace here What do you get? 3 change here right? What will 3 be? See 1 in This. 1 minus what? -2 plus Well, a x = 1, have you added b yet? There. So So what do we have here? A + b will be equal to some? A b = 4 is correct Are not? From there a = 1 then what is it guys? yeah? a = 1 then what? Yes, b = 3 right? Because a = 1 here. Then replace down here b = 3? So I will calculate what is this So 4a minus b will equal 4a here minus b. B I can go now yeah Have we made progress yet? Already. A here 4a - b minus 3 is 1 a and done Remember? There you go You need to remember to do this exercise I just need to remember what is one property? A polynomial function has an extremum at a point. What is the derivative at that point? Equal to 0. Remember? There. Repeat the theory. Sentence Example 13 is the derivative given to you. ready? Let's review the theory. for this teacher Step one is to find the domain. The second step calculates y' right? For y' equal to 0. But this article was given Do you have the derivative ready? Given derivative Already. So do we need to calculate the derivative? anymore? No need. We just need what else to look for? y' = 0. Okay? Already. Any You guys look at me like this. Come on guys let f' = 0 So f' = 0 what do we get? This? This question of my class likes to do quickly or not slow down? Now I will guide you two ways. Way one You can use it subtly as follows. Teacher Quick guide first. How about one? remember this Double root then is not an extremum at the double root is no is not making the sign change correct Are not? So you guys look at sentence f' for me yeah What is this X? X - 1 this x square minus 2 right? This times x 4 - 4 rows of equality right? It is equal to what What is the equality? X average minus 2 times x squared c 2. How do you understand? this place Are not? Because x4 - 4 is a squared minus b b b b squared equals a - b like a + b? So what is this F what? What is this x - 1 times? x squared minus 2 what is this? Have you got the 2nd hat yet? The 2nd power is not correct. Are not? What else, guys? x average 2 is correct? So remember theory for teachers Want to be extreme then y' do Star? miss you teacher X0 is the pole treat Is the extreme point. Must be from point x0 is the extreme point. Right? Then we can deduce that our y' is like any? Y'y x0 what to do? Change what? Change sign. Change the sign right? Want to be the point extreme y'. To be an extreme point, y' must change. sign. Right? So you look at me as follows. X vase - 2 vases have positive? Always positive? Children look, x squared minus 2 squared positive guys? Is the order more than 0? X squared plus 2 positive Not yet? Are you guys happy? Teacher asked the students change sign? Can Yang change sign? anymore? Are not. Because it is positive then how to change the sign again right? Is this positive yet? So these two guys change sign? Change the sign because it always positive right? So it proves that it will can't be extreme right? So there is only one guy x - 1 can change sign only Because it is because so it can be 0 right? Guy which can be equal to 0 can change sign Okay. Understand? Zero is gone positive and negative is not good. So the article this has only one extreme is x = 1 my price is a. Done. Because x = 1 equals 0. Understand? It will be able to change the sign. miss you teacher Only odd multiples. Odd multiples. Odd multiples. Odd set is 1 to 2. This hat 1 hat 2 understand? That is extreme. Ah 1st power 3rd power but sorry teacher. Multiple this odd power 1 power 3 power 5 power 5 power 5 say odd hat, odd hat. That will be it. is extreme treat okay You don't need that. must make table side always. No need to set up side table always Look at the results but this sentence is saying that people laugh are you done yet? Is it necessary to follow the setup method? side table? The super long time I set it to 0 and then check the sign, it's very confusing. time understand? No need to do so. Give me questions 14 and 15. Example 14 This. The function fx has derivative f' equal to this much Not yet? How many extreme points does the function have? how much? You look at me like this. Even numbers are crossed out, guys. This hat even? Because even numbers do not change. correct sign? Always positive. Even hat positive brick The square root is always positive. obliterate. Right? There. So prove it's only x - 2021 to the power of 2021 is odd Not yet? Odd powers can be 0 and it can change sign Remember the odd hat teacher can change sign because odd can be negative or positive. Okay? So we just how many hats Two odd hats. Correct. Right? So it follows that f x has two What is the extreme point? Two extreme points what is x You give this equal to 0 yeah What is X, guys? 2021 or x how much -1. understand? There. Position Stop. It's that easy. My answer is what is this? D. Is it fast? There. I leave Can you give me the even power test? In Even power does not change sign. Ok. That's it. then the remaining odd numbers change sign finished. But there are some articles that need to be changed. there yeah Example sentence 15. Given the function y = fx, The derivative is f'x = 3x² times x - 2 nh x + 2 4 times 4/x squared. Right? This sentence You guys look at me like this. This observation. This sentence is a person's sentence. a little bit mysterious You have to try Try to transform for the teacher as follows. Come on guys look at this First of all. Which one is the double solution, you guys see? like it doesn't exist. Okay? Children Give it to me. This is ok Can you shorten it for me? Then shorten this guy. Throw away two this guy ok So what is f(x) now? What will this be shortened to? Shorten by What? f' nothing more guys yeah. f'(x) is nothing more than that, guys. yeah. You guys take off your sun hats. You guys leave Teacher number 3 go. 3 positive yet? Correct note Are not? Yang left. Yang left. Okay Not yet? We are only there because remember things that are positive do not change sign you guys leave it We only have x - 2 left. how many times yeah? X - 2 times 4 - x squared right? 4 - x is ok guys 4 - x is ok again transform into what remember? Is 2 - x times 2 + x correct? you guys? So what is this? X - 2 times 2 - Are x reversed? Teacher change d sound out is minus x - 2, that is jar. ok understand? 2 - x remove d negative out is -x minus x - 2 x - 2 is x - 2 squared without multiplying 2 together x. Ok. This guy squared it again. Are not? Oh, the dirty test, you guys left it. go on teacher it will not be able to cure can i do it again? So we only have only one extreme is that so f(x) has a unique extremum with a point pole What is extreme value? What is X guys? yeah? What do we do if this place is zero? x = -2. So there is only one extreme point then surely Surely this is the maximum right guys? you? An extreme point is for sure here. maximum Ok what is the answer? Is x = -2 is it b If it's difficult, then where is it? than we have to use computer to click ok? Computer click and review. But This is the fastest way for you. Okay? There you go. After doing this, it will be done in order. After doing this, if we people We want to see if this is a maximum or minimum sign. then do you know how to do it? Then You guys draw me a number line. -2 makes it equal to 0. You choose a price any value For example, the teacher chooses the value -3. Go. -3 teachers come here yin or yang? -3 teachers here. Yin or yang? -3 is this negative or positive? -3 only. This is positive not negative. So it will be like this positive goes up negative goes down. Right. Is it great yet guys? Correct Not yet? Then I have to choose a value. replace f' with that. ok if you want We want to find the specific minimum urine output. yeah D has not gone yet. Then form 4. Form 4 for function graphs. Ok yet? Given the graph of the function Not yet? Then you guys pay attention to this teacher. When People give the graph of the function, then you must be considered as people give graph of function fx or based on what function? f' it very different One sign difference just a comma but very different always. If one is given the graph y = fx ok? Is the graph y = fx ok? Then The whole class just needs to look at the convex point. just give me a tip Concave and convex okay? Convex point concave Convex and concave ok Rough. Convex is the extreme great, concave is extreme pee Remember? You look at the convex point. concave for teacher Look at this. For example here yeah Any. So you guys look at this guy what are the extremes? Look at this. If full fx market Look at this FX, it's protruding above the Do you see the bulge? The peak of the mountain. The peak of the mountain. Polar Peak grand. Okay? Minimal concave. Miss must be concave Not allowed two degrees of suction There. Minimal. Pole pee ok Three extremes. Three minima Not yet? Look at that, you guys. know now There, convex is maximum, concave is minimal. Okay? Well, what if? people give graph f' y = f'x, ok? When we give the graph f'x then what do you have to look at yeah? When we give the graph f', you will not look at the concave and convex points that you must re-sketch. You have to sketch again graph You have to draw and draw again. redraw the variable table heaven Ok redraw the variation table. Redraw closed sky board how know? Oh, you know how to draw which? If f' is greater than 0, f'x is greater than than 0 right? f'x greater than 0 is the forward part on ox is the part on ox. Upper part of ox upper part ox ok f' is greater than 0, the upper part of ox then is the monotonic part is going up. Ok Not yet? f'x less than 0 is the lower part ox is it going down like This. Remember? When people give graphs f' that class I remember having to redraw the board one thank you teacher Okay? There, big f' more than 0 is on ox. f' is less than 0 the lower part of ox then it will go down. Ok. I will demonstrate later. I do Sample sample as follows. Come on, f' as follows This. People give f'x Any. This is the x residence. Ah cut here a b c f'x. Here you go. How many of my class? pole treatment? How many extremes are you? How many poles do you remember? Here is the graph f' guys Do not look at the concave pot point. Not yet? If so, it is wrong. So guys look at the teacher you will rely on the sign for the teacher This. Ox is the line y = 0 right? the you? Did you know? So the part above Ox is the positive part you see here. From negative come along I write here. Sound together. This infinite yang comes from infinite yin. from a to a right? The graph below or above? Below is negative. ok So the I draw it will go down to A. Okay Not yet? That is, what is the graph from A to B? Go up go up. Ok is positive. This is positive. but. ok Continue this. From B to C chart market below? Up and down. So go down Are not? Ok. So come down here. From C to Where is the positive v graph? From C thunderstorm same graph above? Up in the air? This is similar to graph F' above. Are not? Above will have positive sign. Not yet? Below is the negative sign. So the positive will be go up like this There. So we will have how many extremes? This is huge. Here this minimum. This minimum. Maximum. So we will have three extremes. Okay? Those are the three extremes. The function has a minimum up to x = a and x = c. Maximum function t x by what? x = b. ok There you go. When people give the graph f' to the students in class I draw again for you. This is called a table. that's a rough draft. There. Because of me just care if it goes up or down down so I can see the concave point Not yet? Well, this is f(x). There If we have f', we have to redraw f(x) like this. teacher Okay? That's it. Distinguish for teacher Look at the graph of f(x) or f'. FX then concave F' must be re-sketched The variation table above is positive going up to negative. go down there That's it, don't go yet. Ok. Look at example 16. Which gives the function The number y = fx has a graph drawn on the side. Here it is What function graph is this, guys? What function graph? This? fx function right? Then fx. So What point shall we look at? Point concave right? FX is concave. So then guys, the convex point is here. Bulging this is extreme What is the concave point? remember? minimal right? And people ask what The function reaches its maximum at the point any? Oh my gosh this must be huge right? Are not? But where is the bar? The I'm not supposed to say the max is 4 but What is the maximum score, guys? X how much X = -1 right? This is the maximum at what point? Do you understand the point? Is the point about x okay? That's extreme. how much is big X = -1. Okay? There, I will write it again. Answer key my answer is A But you translate give it to me as follows. Function function reach maximum right? Where does it peak? Obtain maximum x = -1. Okay? This. This. In maximum x = -1. Where do you remember from? In x = -1. At x = -1. Well, how big is he, guys? What is the maximum value of Y? This is great. This is the maximum color line. how much Is 4. Understand? Where else? The function has a minimum at the minimum at which point you? At which point does the function reach a minimum? you? The function has a minimum at at what point guys? Remember from now at what x X1. Already then guess how small it is? That's right, guys, which line do you follow? What is the current in y? Minimal. Y minimum must be y line y equal some? y = 0. Is that right? Maximum at x = -1. y maximum is 4. minimum is at x = 1. line then y = 0. Okay? minimum equals 0. There you go. Remember the presentation. 17 this The function fx has a graph as shown. beside. Ask how many poles the function fx has pee right? Here you go. fx chart right? So you look at the pot point. just concave How many minima? you guys give this teacher This is the maximum right? This is minimal. This. This is minimal right? Here it is maximum right? So we will have How many minimum points do you remember? Two points. Two minima correspond to two minimum point right? What is x? and what is x? Minimum point what is x guys? x = 0 and what is x? Correct Already. x = 0 and x = 2. Two minimum points here x = 0 and x = 2 are two poles. pee Because people ask what the score is say is x Okay? Where is the maximum? or at least tell him remember. Ok. So here is my rock. Yes. Not yet. You will know this when you see it. right away Question 18 any. Now, sentence. 18. Given the function y = fx, determine and relate continue on R. Okay? Know the graph of function f(x) as shown. There, f(x) is like picture This. Ask how many poles the function has grand? How many maximum points guys? You guys see the pot point right? One maximum This. Pot up to max yet? How many points? maximum here? Two extreme points right? Then what is it? This? C there Fx graphics remember the teacher's point go up. The pot is extremely hot. Pot up. Ok. Then enter the sentence. 18 which Sentence 19. Look at sentence 19. The ones This post will be mixed up so guys I really have to pay attention to what people give graph of which function. Right? This sentence 19 for function y = fx has continuous derivative on R and the function f' here you see graph f'? Graph f' yet? Already. Which statement is correct? Hey guys, function! whether the number reaches its maximum or not, you will Look at this, you can't guess. where. You have to redraw the chart. Teacher, please show me how to draw quickly. Any First, I will cut the graph at the point any? Where does the fx graph intersect? -2 and how many? 1 right? -2 1 then the Look, from minus infinity to -2 it's at below is above, below is negative it will go down like this you guys remember to follow one this straight line Go down ok? Continue this. From -2 to 1 positive or negative? Positive. Teacher, if you graph f', you must see if it is positive or negative Teacher Duong it is completely above the line y = 0. Do you understand? There. This line y = 0 right? Graph completely above right? So it will go up. Continue this. From 1 to middle extremely. Above or below the graph above or below i cover it. Up or down? Is above positive? It will be like this. There. Okay? FX. Do you understand how to destroy this disaster? Are not? There you go. Sound below then sound is go down from -2 to 1 going up here. 1 to the offering also up here like hey So guys. The function reaches a pole. Is it true or false that x = -1? True or False Are not? where is x = -1? x = -1 is nothing both right? Wrong. The function reaches a minimum at the point x = 1. What does x = 1 mean? Here x = 1 This number is nothing right guys you? x b= -1 is nothing x = 1 is nothing is anything right? Wrong This. The function reaches its minimum at x = -2 true or false guys? That's right. x = -2 here you go guys Downstream is the extreme pee? Minimal yet? Standard with stone what is this C there C right? Children It is very important to look carefully at the graph of which function. yeah Even though it's easy, I'm easy. confused, easily mistaken Okay? The I look at sentence 20 for you. Question 20 for the function The number y = fx defined on r has a function graph The number y = f' is the root. Here is the graph f' yeah Ok. Then which of the following statements is correct? Then now must sketch again for the teacher as follows. The My brother broke the limit very easily. Come on, head. First draw me the line y = 0. Right guys? At what point does it intersect the graph? point here? One point, two points, three this point Right? Remember to close Teacher, follow this order down. I draw it correctly. Okay? Any. So you see the outline as follows. From minus infinity to -2, the graph below is up, down Ok. Ok. From -2 to 0. Above or below? -2 0. You take the pen you cover the pair go in the air? These two parts are above. Are not? So the graph will go up like this This. Do you understand? There you go. From -2 to 2 this. -2 to 2 graphs above in above is positive positive. You mark for The teacher is always negative. This is from 2 to positive infinity graph it is below Are not? F' negative. Negative F' is a regressive Are not? The reversal is going down like this. There you go. Will you know? This this minimum. This is huge. ok So what is rock? This is huge. Draw beautifully. Right? This is the maximum here This is the smallest. So what is rock? what you? The function reaches a maximum at x = 2. Let x = 2 here you guys look X = 2 this. Yes maximum yet? Right. Right. It is what is this? A always. Okay? The function achieved minimum at x = 0. x = 0 is nothing. right? There, the function has three extreme values. wrong it only has a few extremes? 2 only Right? Analyze carefully. Function y = f(x) reaches a maximum at x = square root 2. x = square root 2 This. Apartment 2 here. This line down. There, wrong is nothing right? There, if People give the function f(x) then this is the pole great right? But this f' then Don't do that. Okay? Ok. Wait for the teacher to go wrong. Question 20 is a. Ok. Now, verse 21. Given the function y = fx. Function y = f'x. This. Y = f'x. y = f'x. Okay? There is a graph as shown on the side. Here is the graph y = f'x. f' ok? Read the question carefully, don't make mistakes. Already. In the following statements how many true? Okay? Already. Now You have to sketch the square table again. for teacher Come on guys it's very easy Stop. Whoever is faster will be first. First, which points does it intersect this graph? This? This x1, x2 and x what? X3 right? you guys? This one is from minus infinity to x1 This. The sound is very good. This is so positive. If the negative x1 is infinite, then the graph above is good. below? This gives about this range. positive right? Above that right? Then go up? Go up. Ok. This This. This number line x1 this. X2 this number line I give it then negative v together to x1. Graph it is positive on top. Go up. Next custom This. Ah from x1 to x2 my graph is below above remind you to look. X1 x2 graph at what Down there? So it goes down Are not? Okay? X1 x2 go down yet? From what do you see x2 to x3 above? below? X1 x2 above or below? X3 x2 to x3 above or below? Above the curve above go up. What about X3 to positive infinity? Also on the right note? Both are the same. this yang yang is just yin yang yang x2 x3 to be on top of x3 positive shape v together also on top of v going up like forever ok, you guys write to me right away ok so we will have the result then that is fx always There are three poles on k function y = fx. true or false false this is the maximum right this is very small right this is very small If it's small then it only has a few points. cure Two is fine right? 3 is wrong. Wrong. This is wrong. This. On the function y = fx reaches a minimum at x = 3. Is X3 correct? Reach the minimum from x3. Where is X3? x3 here. The line down is nothing. both right? The line down it does not fall into the wrong extreme. Ok. How small should it be, guys? X some? X2 minimum here. Upstream at x some? X2 yet? Where is X3? X3 X2. Then the function reach minimum to x2 sounds right. Correct So what do we get? Correct Not yet? That's ok. Not yet Go. Sentence two. What answer did the whole class choose? remember? What two? Given the function y = fx with the graph f' market. Here is the graph of f'. Then on The drawn interval then on k the function has how many minimum points Minimal. Ok. Guys, please quickly destroy this for me. First find this teacher intersection. Ok Not yet? Find this intersection y = 0 correct Are not? here Cut at what points This? 1 this 2 3 right guys? Any from negative infinity to this point. Bad points then I consider it x1. Which from negative same point graph below above. This is this sound. Yang Yang Yang. ok About the negative is going down this. Yang Yang Yang is going up. understand? Negative infinity to -4 is negative Not yet? Well, this number is negative. From this number this point to this point is positive go up From this point to this point there is a positive note going up. 4 to yang offering is above. Okay? You have to divide especially what you cut in half below is negative what is above is positive. Okay? Go down and up in one go. So how many is it? very small guys? A minimum is a. Remember? A very small one. Not allowed who look here This is the minimum. This is minimally wrong. Remember? Are not be seen like that. Do you remember when people We punch the graph f' and you have to sketch it. draw for teacher like this. Remember? Then the first table. money breaks you yes