This is not the end of the story. First of all, let's look at the sign of EvdaSX. If the first slope of a point is positive and then negative, or if the first slope is negative and then positive, then neither is increasing nor decreasing. If both sides are positive, then the function is increasing.
Hello my dear students, how are you all? I hope you are all doing well. I am Shashank Singh Rajput, I welcome you all to CWT at 24x7.
So how are you all? Today we are going to do a great one shot on one of the most favorite, most important chapter of math. And that is the none other than application of the derivative.
Meaning most most most important chapter of math. Book number 1 is the last chapter. Before that I have done 5 chapters.
I have done one shot of all the chapters. Relation and function inverse trigonometric function matrices. determinant continuity and different stability by sorry chapter k1 short Adam Katana cleverly garage have jitani me but Tony need a habit of look drooling bar check out girl oh so I'll get session go sure odd data are archie session go sure what character but so finally calculus can the re application of derivative at a or application of derivatives a up to 8 to case based study ga a PS my low key char number castle case based studies a or scale our Apart from this, 2-3 more numbers, that is, near about 7-8 marks is the weightage of this chapter in your calculation.
So, it is a very important chapter. One more thing I will tell you, this chapter is AOD, Application of the Derivative. It is important for the board, if you are giving CBT along with it, or you are giving the paper of JEE, or if you are giving any paper, application of derivative is one of the most favorite chapters of the competition level.
So, if you give any competition, you will get the most weightage chapter and application of the derivative. It means it is one of the highly important chapter. But there is a good news for you.
What is the good news? The good news is that your syllabus has been made very small, which has been running for the last 3 years. This syllabus is very reduced. The application of the derivative has rate of change of quantities, increasing and decreasing function, maxima and minima, first order derivative testing, geometrical representation, second derivative testing.
Simple problem that illustrate basic principle and understanding. understanding of the subject as well as real life situations. So this is your whole syllabus.
Apart from this, there is tangent and normal, there are also theorems, roll theorem, LMVT theorem, mean value theorem, remainder theorem, all theorems are deleted from your syllabus. All these are not in your syllabus. Only this portion is in your syllabus. If I categorically tell you, then this is just AOD's only three portions, one is maximum, minimum, increasing, decreasing and third This is the rate of change of the body. That's all there is.
This chapter becomes a very small chapter at your level. So let's start. Today's session we are going to cover Let's see here quickly. We are going to cover monotonicity of a function.
How to find a monotonic function at any point? How to find it at any interval? What is point of inflection? Next, we will cover maxima minima First order, second order derivative testing, some important techniques about maxima minima stationary point, global maxima, global minima, absolute maxima, absolute minima, and last rate of change of bodies.
So, keep this in mind, rate of change of bodies, I will explain it later, so that you understand the whole chapter, after that we will come to that. So, this is the basic outline, what we are going to cover today. I think all my students are crystal clear.
So, let's start with monotonicity of a function. To understand this, write some basic definitions. First write monotonic function, first write increasing function. Write increasing function.
Finally, what is increasing function? And here write another one, strictly increasing function. You have to understand each word properly. Only then you will understand, otherwise after some time everything will mix up. Strictly increasing function.
I will explain it in detail, don't worry. Just understand it well. Look, what is increasing function?
If any function is increasing like this, means it increases and then becomes constant, then increases, then increases, then increases, then becomes constant, this is called increasing. And what is strictly increasing? If the function is increasing continuously, then we will call it strictly increasing function.
I think all my students are crystal clear. Now, in the same way, write decreasing and strictly decreasing function. Here, decreasing and here strictly decreasing function. All these terms are highly important for you.
The more you understand the basics, the better. Now tell me about this. If it is decreasing, then how?
is decreasing like this. Means it is decreasing, then it is getting constant, then it is decreasing, then it is getting constant. So, this is the decreasing function.
And in strictly decreasing, if it decreases like this, then this is the strictly decreasing function. I think all my students are crystal clear. Now, there is a term neither increasing nor decreasing.
So, here write Nor decreasing. What does this mean? The more you understand all these terms, the more fun you will have in reading.
Neither increasing nor decreasing. What does this mean? This means that if any function increases, then decreases, then increases, increase and decrease, then this is the neither increasing nor decreasing.
For example, y is equal to x square, the graph is like this, then this is the neither increasing nor decreasing function. So, this is the neither increasing nor decreasing function. So, if there is any function like this, then neither an increasing nor decreasing.
Now, I will explain everything in detail, don't worry, I am just explaining you the basic word. Now, after this, add the heading monotonic function. Ok, here add the heading monotonic function.
Sir, what is monotonic function? Look, monotonic function is a function which is either increasing or decreasing. Here write, if a function is either increasing or decreasing, Both of them will be called monotonic functions.
What does it mean to say? It means that we will call this also monotonic. If there is any function of this type. If there is any function of this type, then we will call this also monotonic. Okay?
I think all the students have crystal clear. Now next time, non-monotonic function. Here, put heading non-monotonic function. Non-monotonic. Non-monotonic function, how will we call it?
If a function neither increases nor decreases, simple. So, if a function is neither increasing nor decreasing, so this is the neither increasing nor decreasing non-monotonic function. So, this is the whole story of monotonic city, monotonic word.
If any function is increasing continuously, then we can call it monotonic increasing. If any function is increasing increasing function. If it is decreasing continuously, then we are calling it monotonic decreasing function.
Crystal clear? I think all my students are clear on this. Let's see it again.
What is increasing function? If any function is increasing continuously, then it is increasing. If it is increasing in this way, then it is strictly increasing.
So, if any function is increasing continuously, then it is increasing. If any function is increasing continuously, then it is increasing. If any function is increasing continuously, then it is increasing. If any function is increasing continuously, then it is increasing.
If any function is increasing continuously, then it is strictly increasing. Thank you. Decreasing is strictly decreasing. Neither increasing nor decreasing is strictly decreasing.
Monotonic function is either increasing or decreasing. If it is increasing then it is monotonic increasing. If it is decreasing then it is monotonic decreasing.
Non-monotonic function is decreasing. So, this is the most basic thing. I think all my students have crystal clear.
Now, if you have crystal clear all these things, then let's move ahead to monotonic city. So, monotonic city is a very important function of the monotonic system. So, monotonic city is a monotonic city. Monotonicity of a function at a point. Just put the heading, kids.
Monotonicity of a function at a point. Okay. What does it mean?
It means that if you want to check at any point whether a function is monotonic or not, then what you do is, let's take three points here. Suppose if you want to check here, where, sir? i takes the decouple of two.
Let's say x is equal to a. Let's say you want to check whether this function is increased or decreased. So what we will do is, we will take a bigger point in its neighborhood and a smaller point in its neighborhood.
That means we will take one as a plus h and the other one as a minus h. And this function is y is equal to f of x. I think everyone is clear.
Now here you can see that here who is greater? Since a plus h is greater than what? a. And if a plus h is greater than a, then you can see f of a plus h is greater than what? f of a.
Absolutely correct, sir. Apart from this, here, So, what is A? A is F and you can see that F is F.
So, if we combine both of them, then we have, since a plus h is greater than a and this is greater than a minus h. So the value on their function, f of a plus h, this is greater than f of a and f of a is greater than f of a minus h. If such a case arises, that means if you keep increasing x, and if the value of the function also increases, then in such cases, what will we say?
y is equal to f of x is increasing function. What is this function? This is an increasing function. Where? At x equal to a.
Is it clear? Where is the increasing function? At x equal to a.
I think all the students are crystal clear. Now let's take the opposite case of the same. Suppose we have a function like this. And here we have three points.
Suppose we have to check here at x equal to a. function is increasing or decreasing. So what will you do?
Sir, here we will take one point bigger than this, and one point smaller than this. You know that this is the x axis. And the more you go to the x axis, the value of the point will increase.
That means a plus h will obviously be bigger than a, and a will be bigger than a minus h. So what will we write here? Since the a plus h is greater than a, we will write a plus h is greater This is bigger than a and a is bigger than a minus h.
Then, the value of the function here, a plus h, is the smallest. Where is the value of the function here? It is a minus h. Look at this. What is this function?
y is equal to f of x. height is the biggest, f of a minus h, f of a, f of a plus h. So as you increase x, the value of function is decreasing.
So if you increase x and the value of function decreases, then at this point, what will you say? Sir, y is equal to f of x. is strictly decreasing function. You can write it like this. What kind of function is this?
Sir, this is a decreasing function. Okay, crystal clear? So this is the basic story.
By the way, you don't have to worry much about this. You just have to understand that if any function is increasing, then if we increase x in that, then the value of the function will increase. It's obvious.
Because if this is f, then this height will be f, its height will be f, and its height f. So, as we are increasing x, you can see that the value of the function is increasing. So, if this is the case, then this is the increasing. And if we are increasing x, if the value of the function falls, then this is the decreasing function. So, this is the overall story.
So, let's move ahead and see the graph. Based on the graph, there are many questions in your paper. Why f is equal to x cube is increasing?
Where will it increase? Now, in such questions, you have to draw a graph directly. You don't have to do much math. Why is equal to x? How is this cube graph made?
This way. If it is made this way, then where is it increasing? Sir, it is increasing everywhere. For all x belongs to real number.
It has come, right? So, it is increasing everywhere. Only positive, only negative for all real values of the x. That means, c part is the correct answer.
See, it is increasing from minus infinite to plus infinite everywhere. Next question comes. The value of the a for which fx is equal to a to the power x. Yes.
Where is it increasing on R? For which value of A fx is equal to A to the power x is increasing? I have explained you many times about A to the power x graph.
What kind of graph is A to the power x? What kind of graph is A to the power x? There are two graphs in this. How?
Sir, if a is greater than 1, then this type of graph will be formed. If a is between 0 and 1, then this type of graph will be formed. Now, in the question, it is said that we should increase in R. So, when will it increase?
When a is greater than 1. That means, a is infinite from 1. So, which part is there? C part should be the correct answer. You are seeing all these questions of one number.
In this way, basic questions can be asked. Next question, y is equal to x minus 9 whole square. Sir, where will it increase strictly? In which interval? Now see how this graph is made.
I have also told you about this in shifting of the graph. So how does y is equal to x square graph is made? Sir, this graph is the y is equal to x. square. If we do x minus 9 instead of x, then what will happen?
This graph will shift 9 units here. So, what is this point? Sir, this is the 9 comma 0 and this graph is of x minus 9 whole square.
Now, where is it increasing? You can see that it is increasing here. Yes, it is increasing here.
So, what will we say? Sir, x will be from 9 to infinite. So, if x is from 9 to infinite, then what will it be?
It will increase. So, the correct answer is from 9 to infinite. Okay, is it crystal clear to all my students?
Yes, I think all the students are clear. So, you can see all these questions are based on graphs. You don't have to use a pen anywhere. Next question is, fx is equal to 3x square. So, how does a 3x square graph look like?
Sir, it looks like a x square graph. So, whose graph is this? y is equal to 3x square.
Is it clear? Magnitude of every point is 3 times. The orientation of the graph will be same.
Now it is asking where to increase and decrease. This is obvious. Where is it decreasing?
Sir, it is decreasing from minus infinity to 0. And you can see that it is increasing from 0 to infinity. See the option. Where is the increase? 0 to infinity.
Where is the decrease? Minus infinity to 0. That means C part is the correct answer. Is it looking like this?
You should know the graph. You can easily answer the question. Next question is, fx is equal to x cube plus 1. We have studied very basic in this chapter.
fx is equal to x cube plus 1. Draw its graph. How to draw it? First of all, draw y is equal to x cube. How is y is equal to x cube graph?
Like this. Now when we draw x cube plus 1 graph, then this graph will be drawn on 1 unit. Means, what is this point?
0,1. What is this point? x equals to 0. Now, what is it saying?
Increase in 2 to 3. So, it is obvious that where will be 2? 2 is here and 3 is here. So, it will increase between 2 and 3. Increase at x is equal to 1 only. This is wrong. Only word is wrong.
Decrease will do. It will not decrease anywhere. It will increase everywhere.
Neither increase nor decrease is wrong. Which is the most perfect answer? A part should be the right answer.
Ok, crystal clear for all kids. There should be no problem at all. Ok, very good.
So this was the whole story of basic. Means this was the basic brush up of the kid. Ok, I think all kids are clear.
So if you are clear then tell me in the comment section below that yes sir, everything is clear. Now let's talk about monotonicity in an interval. This is important.
The question of monotonicity is mainly asked from here. Try to understand it. Try to understand it well. I'm doing one thing here. Let's make a beautiful graph like this.
Make a beautiful graph. Now here we have a point, let's take x is equal to a. And we have to find out if this graph is the y is equal to f of x.
We have to find out if this function is increasing on x is equal to a or not. So in this case, how will we solve the question? Look, we will draw tangent at every point. We will draw tangent at every point. Draw tangent at every point like this.
What is tangent? It is basically dy by dx. If any of you don't know, write it down. Tangent is dy by dx.
And dy by dx is also called rate of change. This is also called rate of change. and dy by dx is also called slope. And slope is tan theta. If you have understood this much, then we have done a simple work that we have seen slope at every point of the curve.
If I see the direction of the slope, If I see theta at every point, then theta is less than 90 degrees or not? If we are looking at any point, then theta is less than 90 degrees or not? Absolutely correct sir. So, what can we write here?
Since theta is less than 90 degrees, If theta is less than 90 degrees, then tan theta will be positive or not? Why? Because sir, if we ever, coordinate system, if this is the x, this is the y, in the first quadrant, all trigonometric functions are positive. So, if theta is less than 90 degree, then theta will be here in the first quadrant.
And in the first quadrant, all trigonometric angles are positive. So, tan theta is positive. Now, what is tan theta?
Tan theta is slope. So, slope is positive. And slope is dy by dx. So, dy by dx is positive. So, what we mean to say is that if the slope is positive before any point, and if the slope is positive after that point, then at this point, we will say that the function is increasing.
So here we will write sin. Now, you will understand everything, don't worry. Here, we will write sin of f'x. So, for sin of f'x, what will we do? Here, we will make a real line.
Here, we will write x is equal to a. And here, we will write sin convention carefully. If earlier also slope is positive, later also slope is positive, then here, what will we write? Here, we will write at x is equal to, or write like this, fx is, strictly increasing function i i taxed equals to a per. Clear?
If dy by dx is greater than 0, then fx will increase strictly at any point. Clear? Now let's see the reverse process.
Suppose we have a function like this. And we have to check at any point whether the function is increasing or not. So again, the process is the same. We will draw a slope at every point. We will draw a slope at every point.
In this way, you can draw a slope at every point. So we will draw a slope at every point. And here we will see the direction of the slope. Is the slope positive or negative?
So if we see theta here, then what is theta here? It is greater than 90. Here also theta is greater than 90. Here also theta is greater than 90. So, what is theta happening everywhere? Is it greater than 90 or not?
Tell me. Here also theta is greater than 90. Okay. So, what will we say here?
Here we will say that since theta is greater than 90 degree, then if theta is greater than 90 degree, that means we are talking about this quadrant system. Theta is greater than 90 degree. That means we are talking about the second quadrant. And how is tan in the second quadrant?
It is negative. Why? Because here the sign and the cos are positive. Yeah. Sir, tan theta less than 0. Tan theta if less than 0, then slope less than 0. Slope if less than 0, then dy by dx less than 0. Okay.
What does it mean to say? Sir, if theta is greater than 0, then slope is negative. And if slope is negative, then at this point, this function will strictly decrease.
So here we will write sin of f dash x. Okay, there is no exception in math. There is a reason behind everything. There is nothing that you have to understand.
You have to understand everything here. So this is the x is equal to a. First if the sign of dy by dx is negative, then also if the sign is negative.
Okay, so what will we say here? fx is strictly decreasing function. I think all my students are clear on this.
So, let's move ahead. I am writing a simple concept here. Write here important note.
This is highly important, you just have to keep this in mind. Here write sin. Or here keep in mind, sin or sin. So, there will be 4 cases of sin. So, make this quickly.
So, here sir, x is equal to a. The first case can be that x is equal to a, first also slope positive, later also slope positive. So, here what we will say, fx will be strictly increasing function. at x is equal to a. Okay.
Similarly, if we take x is equal to a, if the slope is negative, and the slope is negative later, then what will we say? fx will be strictly decreasing function at x is equal to a. Okay. What can be the third case?
If this is x is equal to a, First slope is positive, if the slope is negative, then in both cases, neither increasing nor decreasing function. Neither increasing nor decreasing function. So this is the overall of the beta. This is the overall of the beta. First of all, we have to find the sign of f of s.
If the first point of a point is positive, then negative. Or if the first point is negative, then positive. So neither increasing nor decreasing. If both sides are positive, then increasing. If both sides are negative, then decreasing.
It's easy, isn't it? Very easy. Okay?
I think all the students are clear. Now the most important thing here is this, which is asked in your paper, separating the interval of monotonicity. Now let's understand what we'll do here.
Look, here write let we have a function, y is equal to f of x. Okay. Now what we have to do in this? We have to do steps.
Okay. First of all, you do this. Find f dash x. Okay.
What you have to do next? Second thing. First of all, find f dash x. First of all, find f dash x.
Okay. Use. Sine of f dash x. Okay.
And if there is something else, then we have to find it in a different way. Mainly polynomial is asked. If there is trigonometric, then we have to solve direct inequality. Okay. So, sine of f dash x.
So, if you want to see this, then you can do this. The third thing we will do here is, if the first slope is positive, then negative, sorry, if both sides are positive, then function is increasing. If the first slope is negative, then negative, then function is decreasing. This is the whole method.
Don't worry, we will ask the question now, so you will understand. Here is a term called Wavy Kerr method, you must have studied it in class 11th. First of all we factorize any polynomial, make all the factors on real line, start from right side plus plus minus plus minus alternate, after that in interval, sign is positive, we will say this is the strictly increasing and where it is negative, we will say strictly decreasing.
So this is the overall concept. If anyone doesn't know the wavy curve method, then tell us in the comment section below. I will make a 25-minute video on it.
So you guys check it out. So let's see some questions based on this first. Find the interval where it will increase strictly. So you have to tell the interval. Try to understand the question once or twice, you will understand it easily.
So here given f of x, what is given to us? Sir, 7-4x-x square is given. So the first thing we will do is, we will subtract f'x. So what will f'x be? Sir, it will be minus 4 minus 2x.
This will be it. So f'is this one. If we take common of minus 2, then what will happen here?
2 plus x will be f'. How much is it? Minus 2, 2 plus x.
I think everyone is clear. Now here we need for strictly increasing. So the first thing is for strictly increasing function. Write it completely in the board. For strictly increasing, what should happen for strictly increasing?
Sir, dy by dx should be greater than 0. So, it means that this minus 2, 2 plus x written here should be greater than 0. Now, you know that in any inequality, if we multiply by minus, then the sign of inequality turns. It means 2 times. 2 plus x if we do then it will be less than 0. So, 2 is a positive number, we can divide both sides by 2 plus x less than 0. So, what happens to x? Less than minus 2 comes. That means if x is less than minus 2, that means if x belongs to minus infinity to minus 2, then this function will strictly increase.
Then f of x will be strictly increasing function. Okay? In the board, write the entire beta increasing. Clear? Now, suppose if I ask strictly decreasing in this question, then what do we do?
For strictly decreasing, suppose if I ask strictly decreasing, then what do we do? So, strictly decreasing, sir, here f dash x is less than 0 and f dash x is minus 2 times 2 plus x, less than 0. So, here 2 plus x is greater than 0. So, x is greater than minus 2. So, what do we conclude? That means x belongs to minus 2 to infinity. So, here f dash x is less than 0. And f of x, what will it be?
will be strictly decreasing function. What will happen sir? Strictly decreasing function. Is it clear? So, this is the overall concept.
I think all my students are clear. Let's move to the next question. The next question is on your screen.
Sir, the function of values of x for which the function fx says where it will decrease. Okay, let's quickly look at this question. So, here, sir, given f of x, what is given?
fx is equal to 2 plus 3x minus x cube. Right. And, son, we need a decreasing function.
But, whatever you need, the first thing, the first thing you have to do is to get f dash x. So, what will f dash x be? 3 minus 3x square.
That means, sir, 3, 1 minus x square. That means, sir, 3, 1 plus x. 1 minus x is here. Okay, clear? So, if we take minus out of this, then we can write this as minus 3 times x plus 1, x minus 1. We can write f dash x like this.
Okay, clear? Now, here we need decreasing. So, first of all, let's see decreasing.
Let's see decreasing. For Decreasing function. Here I will explain you a small thing. Keep in mind. For a decreasing function.
Okay. For a decreasing function, what should be f dash x? Less than 0. Keep in mind.
If decreasing is here, then you will put equality here. Okay. If it is strictly increasing somewhere, then inequality is greater than.
If it is strictly decreasing somewhere, then inequality is less than. Okay. Clear? And if somewhere it is just saying increase, then you will put inequality greater than or equal to 0. And if somewhere it is just saying decreasing, then you will put inequality less than or equal to 0. Okay? So, this is a basic thing.
Keep one small thing in mind. Subjective question. We have to keep this in mind. Objective questions will be understood by options.
There is not much problem but keep in mind the subjective questions. If somewhere it is written decreasing then inequality will be there. And if somewhere it is written strictly then inequality will not be there. This is a minor thing but important. So from here what will happen?
Minus 3 times x plus 1 x minus 1 less than or equal to 0. So if we divide minus by 3 then what will happen? The sign of inequality will be reversed. After that divide by 3. So the sign of inequality will be same.
Because 3 is a positive number. Now what we will do? Critical point, we will apply the wavy curve method.
What does wavy curve method say? Using wavy curve method. What does wavy curve method say?
Critical point is x equal to minus 1 and x equal to plus 1. So, here is plus and here is minus and here is plus. Now write the interval in which plus is there. So, x belongs to minus infinity to minus 1. Now see here inequality is there. So, we will put a closed interval on minus 1. So, we will write the interval in which x is there.
and always put open interval on plus or minus infinity. So this is the first portion of the question. The second portion, I am telling you that there is no second portion.
Suppose if we have to write for increasing function, suppose if this question is increasing, then what we do? For increasing function, what happens for increasing? f dash x is equal to minus 3x minus 1x plus 1 greater than 0. Okay.
So, if we multiply by minus, then what happens? x minus 1x plus 1 less than or equal to 0. Okay, so here we will apply the Wehweker method again. So this is the minus 1 and this one is the plus 1. By the way, you need to do it only once if you are asked the same question.
So here, from where do we take the interval? x belongs to minus 1 to 1. Okay, clear? So for decreasing, x will come from here to here.
And for increasing, if it was in this question, then for increasing, we would be doing it this way. I think all my children are crystal clear. Come, let's see one more question.
You will get more clarity. Don't worry. Next question is on your screen. Next question is on your screen. Here it is.
Look what is written. Show that the function. This function is long and wide. Function this one, it is strictly decreasing from pi by 2 to pi.
This is the question. Let's see what the question is saying. So, what is fx equals to? This is given as 16 sin x upon, the first thing you have to do is to get f'x.
Come on, let's do it. So, what will f'x be? Here we will use u by v method of division.
So this will be 4 plus cos x whole square. So first we will do 16 cos x. Did I write it correctly?
Absolutely correct. Minus. Now we will keep 16 sin x as it is. We will do the derivative of the lower one, 4 to 0, what will be cos x? It will be minus sin x.
And x will be minus 1. Okay? Is equal to. Do it properly, with confidence. Okay? There is a small mistake here.
I didn't write 4 plus cos x here, right? Yes. So here 4 plus cos x should also be written.
16 cos x and this is 4 plus cos x as it is minus 16 sin x as it is and 4 plus cos x minus sin x and minus 1. This is the thing. So, now let us simplify it from here. So, it is 64. cos x plus 16 cos square x minus minus plus 16 sin square x, right, minus, here it will be 4 plus cos x whole square, completely divided by 4 plus cos x whole square, this is what we have, f dash x, right, now let's simplify this, cos square x plus sin square x, so this will be 1, from here 16, Now, let's open it. So, from here, F dash x is equal to 4 plus cos x whole square.
How much is it? 64 cos x plus 16 64 cos x plus 16 minus 4 plus cos x whole square. We have to open it and write it.
plus cos square x, 4 plus, plus 8 cos x. Okay, clear? So this is the thing here.
So from here, 16 to 16 is cancelled. Right? If we subtract 8 from 64, then 4 and 4 will be 56x cos x divided by 4 plus cos x whole square f dash x. If I simplify this a little more, then this cos x will be common and here 56 minus cos x. So this is the thing.
Now understand that the whole square of 4 plus cos x will always be greater than 0. If we add 4, then obviously it will be greater than 0. Cos x always lies between minus 1 and 1. So if we add 56, then this term will always be positive. Yes or no? What does it mean to say?
This whole term is always positive. That means, the sin of f'x depends on what? Only on cos x. So write from here, since sin of f'x only depends on sin of cos x. Yes or no?
Okay. So, if I explain you with its graph, then how does the graph of cos x become? It becomes like this.
Yes, right? How does the graph of cos x become? Like this. Okay.
So, this is 0. This is the pi by 2. And here pi is somewhere. Right? So, you can see that when x is between 0 to pi by 2, right? Look.
If x belongs to 0 to pi by 2. So, here, what is cos x? Cos x is positive. If cos x is positive, then what will be f'x?
It will be positive. And if f'x is positive, then what will be fx? Increasing function.
Similarly, if x belongs to pi by 2 to pi, So, here cos x is below, that means cos x will be less than 0. That means if cos x is less than 0, then f dash x is less than 0. So, what function will be fx? Decreasing function will be there. What function? Decreasing function. Okay, is it clear to all my children?
Yes, it is. So, this is the thing here. I think all the children are clearly clear. What is being done here? Yes, this was to prove that it will be decreasing from pi by 2 to pi.
So, you can see this. Now, here... So, here you will get a question that sir, will these intervals be the ones where this will increase or decrease? It is not like that, there will be many intervals, I have written only one interval, this is in fact from minus pi by 2 to pi union 3 pi by 2 to 5 pi by 2, all the odd multiples of pi by 2 interval like pi by 2 minus pi by 2 to pi by 2. Thank you. 3 pi by 2 to 5 pi by 2 whatever odd multiple of pi by 2 interval is being made there it is getting positive and this pi by 2 to 3 pi by 2 here 5 pi by 2 to 7 pi by 2 whatever is being made here it is getting decreased so this is what we had to prove I think all the students are clear hence proved Okay, clear?
Okay, if any child does not understand this graph, then I will tell you one more method of quadrant system. So, if you do not understand this graph, then you can take help of quadrant system. Sir, in quadrant system, all the signs are positive. If any theta is between 0 to pi by 2, then you can see that cos theta is positive and if any theta is between pi by 2 to pi then you can see that cos theta is less than zero so you can show examiner in any way whether through quadrant system or graph both are equally important means both are equally correct there is no problem many people say that don't use graph in board but in board you are not taught graph that's why everyone says no otherwise math without graph is nothing thing.
You have nothing. Do you understand? So, take good care of all my children.
Okay, is everyone clear? Yes or no? Look quickly. Okay sir, everything is crystal clear.
Let's quickly do a question. It's a very good question. All my students, please look at it.
But sir, interval in which the function, in which this function is decreasing, what is it? It's polynomial. First of all, we need to find f dash x, use the wavy curve method, and then check the sign convention. So, here fx is given to us, so what will be f'x? 3, 2, 6x square minus 6x minus 12. So, take 6 as common.
This will be x square minus x minus 2. So, what will happen from here? 6. x square minus 2x plus x minus 2. Solve it. This will be x minus 2x plus 1. f dash x will be this. Yes or no?
Now, it is saying for decreasing, we have to see decreasing. So, whether we want to see increasing or decreasing, you will always use the way we did method. Using Wavy curve method. So, what does wavy curve method say?
Sir, if we have here, write whatever critical point is here, x is equal to minus 1, x is equal to 2. Start plus from here, alternate plus minus plus. So, where the sign is positive in the interval, it is increasing, where it is negative, it is decreasing. So, it means, since x belongs to minus infinite to minus 1, union to select a infinite yeah but I would ask scour sorry yeah but I would ask greater than or equal to zero or a unique sir FX casa hora FX will be increasing or a garix minus one select two kbch man So, here f dash x will be less than 0. So, here fx will be decreasing function. Okay.
So, this is the thing. So, what is the correct answer of decreasing? Minus 1 to 2 should be the right answer. Okay, everyone is clear.
There should be no problem. There should be no problem at all. Okay?
I think all my students are clear. Let's move on to the next question. Alright, come on. Let's move on to the next question. Here it is on your screen.
In this, we have to tell where it will decrease. So, the first thing we will do is to find f of x. So, what is f of x?
1 by 12. This is 12x cube plus 12x square minus 12 into 2. How much will this be? 24x. Alright.
So, take 12 from the top. Below is 12. This is x cube plus x square minus 2. So, minus 2x, so 12 to 12 cancel, yoga x, x square plus x minus cut 2. So, we did its factorization, the same factorization will be done here also. means x plus 2 x minus 1. Okay. So here you can write this x as x minus 0. So again here we will use the Wavy Kerr method.
Use it. Make a real line. So x is equal to minus 2. x is equal to 0, x is equal to plus 1. Write all the critical points.
Here x is equal to minus 2, x is equal to plus 1. So, from here start plus, minus, plus, minus. In the interval where there is minus, it will decrease. So, write here since x belongs to minus infinity to minus 2. Look, decrease and increase is union from 0 to 1. Here, f'x is less than 0. So, here, fx will function decreasing. And if this x is between minus 2 to 0, Union is from 1 to infinity.
What is f'6 doing here? Positive. So, how will fx be here? Increasing function. Okay.
So, this is the thing. So, we have asked decreasing. So, minus infinity to minus 2. Union 0 to 1. So, this should be correct answer. Now, many of you might have a question that this is closed but option is open. So, never confuse about this.
You will see this many times, you will be asked about increasing and decreasing but in optional questions, no one puts a close bracket. Mainly, increasing and decreasing means strictly increasing and strictly decreasing. You can see this according to the options in objective questions. But when you will ask a subjective question, you will pay attention to inequality, so you will put inequality in such questions. Is it clear to all?
I think everyone is clear. So, after this, we will see the next segment, that is the maxima and minima, which is the second portion of the application of derivative. I mean, you can assume that these two questions are the questions. Most of the questions are on this, case-based study questions are also there.
We will see the case-based study questions separately, but the conceptual portion is mainly these two. Now, let's see local maxima, local minima at a point. It has one more name, local maxima, local minima at a point.
Its name is Relative Minima Relative Minima. Relative Minima Relative Minima. Relative Minima.
So, if these two names are written somewhere, keep these two things in mind. Now let's understand what this definition says. It's easy, nothing complicated.
Let's assume we have a situation like this. Let's take one case here and the other case here. Now, here we have a point.
Let's say this point is the x is equal to a. Let's take this point as x is equal to a. And now if we want to know the maximum or minimum value of this point, then what we will do is the same thing we were doing till now.
Here in its neighborhood, we will take two points. One point will be bigger and one point will be smaller. Understand this carefully. What we did is, we took one point bigger than this and one point smaller than this. Now, what will we give as the conclusion here?
Understand this carefully. We are giving the conclusion here, since The a plus h will be greater than a, but the f of a is the largest. So here the value of f of a is greater than f of a plus h. Since the a is greater than a minus h, but the value of f of a is still greater than f of a minus h.
It means that if the value at any point is greater than the value around it, see, if the value at a is greater than the value around it, then what we will say at x is equal to a, this is the point of local maxima. So what will happen at x is equal to a? Point of local maxima.
Please understand this carefully. Point of local maxima. Clear? Yes, brother. In the same way, let's take two points here.
Let's say we have to tell here, x is equal to a. So here we will take one point bigger than this and one point smaller than this. This is the a plus h and this is a minus h. Now see what is happening here. A plus H will be bigger than A But the value of F of A will be smaller than F of A plus H You can see the height Since A will be bigger than A minus H But the value of F of A will be smaller than F of A minus H Clear?
So here you can see that if the value of any point is small, then we will call this point as point of minima. So what will happen to x equal to a? Point of local minima. Is it clear to all my students? Yes or no?
So this point will be point of local minima. Is everyone clear? So this is the thing. I think everyone is crystal clear. Let me concide it again.
Why is its name relative maxima and relative minima? Because it is the biggest value in comparison of its surrounding values. Whenever we compare with someone, the term relative is used. Means the biggest value is in relative of this point.
Then this point is the point of local maxima. The value at this point is small in its relative. That is why this is the point of local minimum. I think all my students should be crystal clear.
So, let's quickly see a question based on this. The first question on your screen is, find the local minimum value of the function. So, local minimum value where x is real number, so again we will draw its graph here.
How the graph of y is equal to mod x plus 2 will be made? This is the 0, 2. So, where is its local minimum value? At x is equal to 0. So, if y is equal to mod x plus 2, its local minimum value What is the local minimum value of this? The local minimum value is 2. Where is it?
At ix is equal to 0. At ix is equal to 0, what is the value? What do we tell? The value is basically fx.
So, the value of fx is 2. The smallest value is 2. 2 to the infinite. But, if you ever ask which x is it at, then x is equal to 0. So, here we have local minimum value. minimum value.
So, what is the value? 2 is the correct answer. Next question comes, minimum value of this one.
You see, it is saying minimum value. So, what will we do sir? Again, it is a graph of modulus function. How will it be?
x is equal to 1 by 2, but it will be non-differential in this way. So, it will be 1 by 2, 0. So, what is its local minimum value? Here, obviously, the value of y is 0. So, the local minimum value is 0. Where?
At x is equal to 1 by 2. You can understand it like this, the value we tell is the value of y. On which x? That x is equal to 1 by 2. Okay, dear, is it clear? I think all my children are clear in this way.
Now, make a note here and write. Note, neither maxima nor minima. Neither maxima nor minima.
What does neither maxima nor minima mean? Look, this means that, If any function is of this type, that it keeps on increasing or if any function is of this type, that it keeps on decreasing, then you can see that here is a maximum and minimum point. Sir, there is no point of maxima and minima here. Is it coming?
It is not coming, right? It is continuously increasing and decreasing. That means both are cases. Neither maxima nor minima cases.
You can also say that sir, if any function is continuously increasing or decreasing, how will both functions be? Neither maxima nor minima cases. Okay, clear?
I think all my students are clear. Now, the next concept is extrema. Extrema.
What is extrema? Extrema means either local maxima or local minima. Both are called extrema. Now understand the concept of extrema. Suppose we have a curve like this.
Make some points here. Give name A, B, C, D, E. If we draw slope here, So, you see here, whose slope is parallel to this? This is the x-axis.
Yes, isn't it? So, here obviously, here sir, theta will be 0, theta will be 0, so sir, tan theta will be 0, tan theta will be 0. So, here dy by dx will be 0. Why? Because slope will be 0. You can understand that if any line is parallel to x-axis, then here slope will be 0. Similarly, here also slope will be 0. Similarly, here also slope will be 0. At this point also slope will be 0 and here also slope will be 0. If we want to write, we can write f'b equal to 0, f'c equal to 0, f'd equal to 0. What can we write here? f dash e equal to zero. What does this mean?
If we have any function where this function rotates or where its maximum value is or its minimum value is here its f dash x is zero. So here you make this point and write at the extrema What is dy by dx? It is 0. Is it clear? What is dy by dx on extrema? It is 0. Or write it like this, if fx Attains extrema i at x is equal to a, then f dash a will be equal to 0. Okay, clear?
Let's quickly do this question. See what is written here, we have a function here, it says attains local maximum value at i at x is equal to 1, so you have to tell me what is the value of a. Okay?
So, what will we say? Since f of x attains local maximum value local maximum value at x is equal to 1. So, what will we say here? F dash 1 is equal to 0. So, first of all, you remove f dash x. So, what will f dash x be?
4x cube minus 124x plus a. So, if we make f dash 1 equal to 0, then we will put 1 everywhere. So, 4 into 1 minus 124 plus a is equal to 0. So, how much value of a has come? 120. Okay. So, 120 should be the correct answer.
Okay. Is it clear to all my children? Yes or no? Is this concept clear to all my children? Very good.
Great error. Come to the next question. I think this question should be done by you all. Read the question on this base. The fx is equal to this one.
It is saying that its maximum value is x is equal to minus 1. So, you have to tell the value of the k. Directly, f dash minus 1, you do 0. So, first of all, remove f dash x. This will be 3x square minus k. So, what will be f dash minus 1?
This will be 3 minus k equal to 0. So, what is the value of k? 3 is here. Okay.
It is an easy concept. Sir, at any point, if it is extrema, then f dash x will be equal to 0. Extrema means either local maxima or local minima. Okay, clear? Very good. Now, next concept is the critical point.
Okay, critical point. Finally, what is critical point? Try to understand critical point.
Okay. Write here. We will write the definition in a while.
First, you understand its concept of critical point. Suppose we have any function. Okay.
If we have any function. Okay. Write all these points here, A, B, C, D. Okay.
A, B, C, D, E. So, what will happen here? Sir, here F dash A will be equal to 0. Here what will happen? F dash B will be equal to 0. Here, what will happen?
Sir, f dash c is equal to 0. What will happen here? Here, there is a sharp turn. That means f dash d is equal to does not exist. Okay. What will happen to f dash e?
Sir, does not exist. Okay. So, if we have any function, if f dash x is 0 or does not exist, we call both of these things critical point. Okay. So, we call both cases critical point.
So, write the definition here. For, write, if we have a function, y is equal to f of x, then the critical point of y is equal to f of x will be at dy by dx is equal to 0. dy by dx is equal to does not exist. Okay.
So, let's take both these cases as critical points. Where dy by dx is equal to 0 or does not exist. Okay. So, let's see its example quickly.
Let's assume y is equal to mod x. Okay. Its find critical point.
Find critical point. So how will you tell? Sir, here if you draw y is equal to mod x.
Right. So here you can see sharp turn. So what happens when x is equal to 0?
x is equal to 0. Critical point. Right. Suppose y is equal to mod x minus 1. Then how will the graph of mod x minus 1 be made?
It will be made in this way, 1, 0. So what is the critical point? x is equal to 1. Okay, clear? Suppose y is equal to x square minus 1. How will its graph be made?
X square's graph is from here, so x square minus 1's unit is from below. So, where has this critical point been? X is equal to 0. Right. X is equal to 0, what has happened? Critical point has been made.
Clear? So, let's see a question on this. Son, this is the critical point of the quantum window. So, if fx is given to you, then we have to tell the critical point. So, for critical point, What will you say?
Sir, f dash x is equal to 0. That means 2x minus 2 is equal to 0. That means 2 times x minus 1 is equal to 0. That means x is equal to what? Cp. x is equal to 1. Is it clear to all?
Yes or no? Tell me quickly. Come to the next question quickly.
Here we have given this point, this curve fx, so we are being called its critical point. Okay. So, for critical point, what we will do for critical point, sir, f dash x equal to 0. So, this will be 3x square plus 2x plus 1 is equal to 0. This will be done.
So here if you see its d, db square minus 4ac is there. So from here b square is 2 square minus 4, c value is 1. So what happens? 4 minus 12, means minus.
And this is it............................. f dash x will be greater than 0. Okay. That means, f dash x cannot be equal to 0. It cannot be 0. Because if d less than 0, then it will never have any real root.
If it does not have any real root, then it will not have a critical point. Okay. So, here we will say, no critical point.
Here, there will be no critical point. Why? Because it will not have any factorization, if d less than 0. Okay. So, what will we say, sir? Do not exist is the correct answer.
Okay. Clear? Very good.
Now the most useful thing starts from here, from where there is first order derivative testing. As we had studied in monotonicity, in increasing and decreasing, the same is here also. Suppose we have this function and at this point we have to find out if it is maxima or minima on x is equal to a.
So how will we find out? We will write x is equal to a here. So, the process is that you first see the direction of the slope and then see the direction of the slope. So, what is happening here?
You see, theta is less than 90 degree. Here, first theta is less than 90. First theta is less than 90. Here, theta is 0. Then, what is happening? Theta is greater than 90. Yes or no? Tell me. So clearly you can understand from here that if the slope is positive before any point, sorry, if theta is less than 90 degree, then what will happen here?
f'x will be positive. If theta is equal to 0, then f'x will be equal to 0. If theta is greater than 90, then what will happen here? f'x will be less than 0. That means if the slope is positive before, then the slope is negative, and if the slope is 0 in the middle, then what will happen in this case?
It will become point of local maxima. So, if you have a point of local maxima, Here you write sin. What will be the sign of? Suppose here x is equal to a.
If the slope is positive before this, and if the slope is negative later, then you think that if the slope is positive, then the rate of change will increase, and then the slope will decrease. That means there will be some point where its maximum value will be. So what will be x is equal to a? point of local maxima and from here, what will be f of a? This will be maximum value.
Okay. So, if you ask for maximum value, then put the value of x in the function. Okay. And what will be x is equal to a?
Point of local maximum F a will be maximum value. In the same way, if we have a graph like this, and if we ask the sign convention here, if we draw a tangent everywhere, then the tangent sign will be negative like this. Then the tangent will be positive here. Yes or no?
So, here our slope is greater than 90 degree, here theta is greater than 90, here theta is equal to 0, here theta is less than 90, theta is less than 90 degree. So, here if theta is greater than 90 degree, then here f dash x is equal to 0. Now, what will be your rate? Here, sir, f dash x greater than 0. Why?
Because theta is less than 90. So, if we write sin of f dash x, then how will we write sin of f dash x? Here, critical point will be written as x is equal to a. First slope negative, then slope positive.
So, you think that if slope negative means function has fallen, then slope positive means function has taken place. That means there must be some point in between where function has the smallest value. So, here x is equal to a will be point of local minima. And what will be fA?
It will be fA, sir, its minimum value. Okay, is it crystal clear to all my students? Yes, right?
So this is the thing, man. Okay. So now if we write its First of all, if we have any point, sin of f dash x, First slope is positive, then if slope is negative, then what will happen?
Point of maxima. What will happen? Max value.
Clear? Similarly, the second case can be that if we have any point, first slope is negative and then slope is positive, here x is equal to a, point of minima, fa will be minimum value. Next, if we have slope is positive first and then also slope is positive, then it will be continuously increasing function. And if such a case is there, first slope is negative and then also negative, then it is continuously decreasing.
In both these cases, we will not be able to find the maxima nor minima bhi kaya sakta hai. Okay. Neither maxima nor minima.
Isko hum increasing or isko hum decreasing bhi kaya sakta hai. In terms of the increasing and decreasing. So, this is the overall nichod yaar. I think sabhi bachko ko clear hai.
Okay. Ab zara step to find out maxima and minima dekta hai. Akhir kar yaha par step kya hai.
Bita in steps ko hum first order derivative testing bhi kaya sakta hai. Okay. To isko bhi note down kar lena.
To yaha par leke lena. First order. First order derivative testing. First order derivative testing, what is it called?
Note down its rule. What is the first work? First work is to find critical point.
find critical points take a say sir dy by dx for our zero curdo yeah does not exist character take us to about gaga na us ke bad using sign convention take using sign conventions agar kissy be point per palace slope negative body positive So, this is the point of minima. At any point, first slope is positive, then negative, so this is the point of maxima. So, this is the method of doing this. This is the method of doing this. This is the method of first order derivative testing.
If we have any point, any curve, write it here. Let we have A function y is equal to f of x in the defined domain, in or down, in the defined domain. So, first of all, critical point, check the sign convention like this.
Let's do the base question on this, you will understand. Like, here we have a function written. We have to know how many local maxima and how many local minima are there in this. So, what we will do is, first of all, we will find the critical point. So, for critical point, don't write CP in paper, write critical point in paper.
So, here we will make f'x equal to 0. f'x will be 3 to the power 6x square minus 6x minus 12 is equal to 0. Right. So, from here what will happen? x square minus x minus 2 is equal to 0. This is x square minus 2x.
So, this is x minus 2 x plus 1 is equal to 0. Now, what we will do here is, using first order derivative testing, First order derivative testing we will use. So, this is x is equal to minus 1. Here, x is equal to 2. So, plus minus plus. Okay.
So, where the slope is positive, here the function is not. increase correct our negative over decrease correct so since X belongs to minus infinite to minus 1 union to select a infinite yeah but I said success are greater than 0 Oh sorry, we will not do it like this. We are looking at the maxima and minima here. So, look here, what is happening here?
Here the slope of the function is positive to negative, that is, the function increases and then falls from here. So, what is x is equal to minus 1 here? Point of maxima. What is x is equal to 2 here? Point of minima.
So, this is the final answer. One maximum and one minimum is the correct answer. Suppose, I ask you the maximum value. Suppose, I ask you the maximum value. Tell me the maximum value.
What will you say? F of minus 1. Suppose, I ask you the minimum value. If I ask you the minimum value, it would be F of 2. So, this is how we can calculate the value of f of minus 1. Is everyone clear?
Yes or no? Okay. Next, we'll do some further questions.
First of all, we'll read about second order derivative testing. Okay. So, what is second order derivative testing?
What is seen? We have a polynomial. If we have a polynomial, you can factorize it and see the sign convention from here. By doing plus, minus, plus and all.
Okay. But it won't always be like this. Sometimes it can happen that you don't see the sign convention How will you see sign conventions in this? You won't be able to see it. So, in such a place, we use double derivative testing.
We use second order derivative testing. Generally, you will see this in many places that if first order derivative testing fails somewhere, then second order derivative testing is used there. Which is wrong.
It is not like that. You can use second order derivative testing anywhere. It is not like that. It is necessary to fail first order derivative testing. No.
If it is not a fail, then you can use it. Okay. So, note down its rule.
Let we have a function y is equal to f of x in the defined domain. Okay. So, what is its rule?
So, first thing is to find critical point. How? As usual, f dash x is equal to 0 or it is a degenetic gesture.
Second thing is to find f double dash x or d2y by dx square. you will get d2y by dx square. Now, here comes the most important point. The sign of d2y by dx square is negative at any critical point.
Suppose at x is equal to p. Then x is equal to p will be the point of maxima. Remember that the sign of double derivative for maxima should be negative at any critical point.
Or if the sign of double derivative d2i by dx square is greater than 0 at any critical point at x is equal to q, then x is equal to q will be point of minima. Okay, so if the sign of double derivative is positive, then this is the point of minima. If the sign of double derivative is negative, then this is the point of maxima.
Okay, is it clear to all the students? Yes, come quickly and see. I think all my students are clear.
Now, this is the last question. This is the question we were asking. Now let's do this with double derivative testing. So the first thing we did was, we took f'x out. This was 6x squared minus 6x minus 12. So from here 6x squared minus x minus 2. So for critical point, f dash x to the power 0, so from here x square minus x minus 2 is equal to 0. So this will be x square minus 2x plus x minus 2 is equal to 0. So this will be x minus 2x plus 1. So critical point is x is equal to 2 and x is equal to minus 1. Now to check the maximum, we will take out its double derivative.
So take out d2i by dx square. So what happens to d2i by dx square? It becomes 2x minus 1. Now look at the sign on both the critical points. So d2i by dx square. I add x is equal to 2. Remove it.
This is 4 minus 1 is equal to 3. That means it is positive. That means x is equal to 2. This is the point of minima. Right? Similarly, d2i by dx square x is equal to 2 minus 1. Remove it. Minus 2 is equal to minus 3, means less than this one.
So, what happens when x is equal to minus 1? Point of maxima. Okay?
So, this thing came from that too. This thing came from first order derivative testing too. This thing came from second order derivative testing too. Yes, isn't it?
Is it clear to all children? So, this is the thing. I think all my children are clear. Okay?
Great, sir. So, today we'll ask a question. One second. Hmm.
So, today we'll ask a question. We have to tell the maximum value of fx is equal to this one. Okay?
Now, we have to tell the maximum value of this question. So, here we can write, to find max value of fx is equal to 1 upon 4x square plus 2x plus 1. So, here we should find, we should find, we can find the minimum value of this. We can find the minimum value of this.
4x square plus 2x plus 1. Yes or no? If we want to get its maximum value, then what do we do to the denominator? We do minimum.
That means if we get its minimum value, then overall what will happen to it? It will get its maximum value. So, let's assume that G is equal to 4x square plus 2x plus 1. We will remove the minimum value of this and put the minimum value here. We will get the maximum value, the basic funda.
So, from here, what will be G-x? 4 to the 8x plus 2. So, let's remove its critical point. For critical point, G-x is equal to 0. Any 8x plus 2 is equal to 0. x is equal to 2, minus 1 by 4. Okay, clear?
Now, this minus 1 by 4, it can be maximum point, it can be minimum point. So, for this, what we do is, you check g double dash x. So, what will be g double dash x? It is d2y by dx square. So, this is...
So, this is always positive. Meaning, if you keep any critical point here, this will always be positive. So, this at x is equal to minus 1 by 4 will also be positive.
Because of this, x is equal to minus 1 by 4. What has happened? This is the point of what? This is the point of minima.
Yes, right? Why? Because g double dash x is greater than 0. So, x is equal to minus 1 by 4, what will be the point of minimum?
Now, what will be the minimum value? So, the minimum value will be g minus 1 by 4. So, where will g minus 1 by 4 come from? From here. Now, solve this g of minus 1 by 4. So, g minus 1 by 4, this will be 4 into minus 1 by 4 whole square plus 2 minus 1 by 4 plus 1. So, 1 by 4 square 4 will be cut, so this will be 1 by 4 minus 2 by 4 plus 1. Okay? So, this will be, minus 1 by 4 plus 1, which means, subtract from 4, this will be 3 by 4. Okay?
So, this means, the smallest value of G, how much is this? 3 by 4. Clear? So, what do we have to tell?
We have its minimum value, we have to get its maximum value. So, what will be its maximum value? Just substitute this thing. So, max value of G.
f of x, what will be this? 1 upon g minimum, so that means 1 upon, what is its minimum value? 3 by 4, so this means 4 by 3 should be the correct answer, so 4 by 3 is the right answer. You can easily solve these problems in the future. But in the board method, you will have to do it like this.
I will not tell you any shortcut in the board method. There is a shortcut for all these. You can easily answer this question in a few seconds without doing so much.
But I will not tell you. The reason behind it is that it will not be in your syllabus. You will have to do it like this at the board level. Okay, is it clear to all the students?
Yes, it is. Okay, great. Let's do one more question. It's a good question.
Generally, many students make mistakes in this question. It says, maximum slope of the curve. Keep in mind, here we have to maximize the slope.
We have a curve. We have to find the maximum value of its slope. Okay.
So, given y is equal to minus x cube Plus 3 square plus 9x minus 27. Okay. Now what will be its slope? Its slope will be dy by dx.
So this will be minus 3x square. 3 2ja 6x plus 9. Okay. This is the slope. So, let's take the slope. Let gx is equal to minus 3x square plus 6x plus 9. Now, the question is that you tell the maximum value of the slope.
So, we have to find we have to find Max value of G. So how will we do it? As usual, the same thing.
First of all, we will do G-X, critical point. We will see the maximum and minimum, at which point it will come, after that we will put it. So what will we do from here? So from here, G-X is taken out for critical point. So from here, G-X will be equal to just 6, 6. plus 6, equal to 0. So, from here, x is equal to 1. Now, is x equal to 1 the point of maxima or minima?
Then, find its g double dash x. To know this thing. So, this is minus 6, which means less than 0. At any critical point, what is its g double dash x?
Less than 0. Because of this, what is x equal to 1? is point of maxima. So, if this is point of maxima, then what will be its maximum value? So, what will be its max value? Max value, this will be g of 1. So, here, take 1 and put it.
So, g1 will be the maximum value. So, it will be minus 3 plus 6 plus 9. So, subtract 3 from 15, that means, how much? Sir, 12 has come.
So, 12 should be the correct answer. Okay, is it clear? This question is a good question.
Generally, children make mistakes in this question. And where do they make mistakes? You know, they make mistakes here.
You don't give the maximum value of this. You start to find the wrong one. Here it is asking the maximum value of slope. So first of all you find the slope and then you have to apply this maxima minima concept on the slope.
So this is the difference. Where generally kids make mistakes. Clear? Very good.
The last concept of maxima minima. Global maxima, global minima. Global maxima, global minima has one more name. This is called absolute maxima or absolute minima. Okay.
There is another name for this. Greatest value in a domain or least value. in a domain. Greatest value in a domain or least value in the domain. So, these are all called global maxima, global minima, local, this one.
So, first of all, we understand a very good thing with a lot of depth. See, make a good curve as I am making. make a curve like this here are many points represent all points make all points ok name of points a b c d E, F, G, H, I, J, K. These are all the points.
Now, it is obvious that there are many points here. So, if we draw a slope here, then the slope will be zero at all these points. And if the slope is zero at any point, then we call that point extrema.
That means, either local maxima or local minima will be there. So, here you can see that the slope is 0 on all these points. That means, here is the maxima and minima. Now, let's understand.
Understand carefully. All these points, which are maxima, What will we call all these? We will call them local maxima.
That is, a, c, e, g, i, k, all of these are local maxima. So write this down. a, a, b, c, a, c, e, g, i, and k.
What will we call this? We will call it local maxima. The minimums are called local minima.
B, D, F, H, J are all local minima. Now, the biggest value in all of these, where is the biggest value? It is at K. So, what will we call K?
We will call it global maxima. That means, the biggest maxima in all of these maxima, that is global maxima. In all of these local minima, where the smallest value is, what will we call it?
Global minima. That means, what will we call J? Global minima.
Is it easy or not? Right? That means in your mind, what I mean to say is that if we have any function, in this function, local maxima can be more than 1. Absolutely right, sir.
Local minima can also be more than 1. Absolutely right. But the global maxima will always be the same. The global minima will always be the same.
This is something that everyone should keep in mind. Basically, global maxima is nothing but the largest maxima among all local maxima, that is the global maxima. Among all local minima, the smallest value is the global minima. That is why it is called absolute maxima and absolute minima.
Because absolute means the largest of all. Absolute. That's why it's called greatest value in the domain. The biggest value in a domain is called global maxima.
The smallest value in a domain is called global minima. Is it clear to everyone? Yes or no?
So this is the thing. Now note down its procedure. Solving the rule of this, note down that.
Let we have a function y is equal to f of x in the defined domain, in the defined domain, take closed interval a to b. Assume that this is given, the interval also. So, what is the rule here?
The first thing is to find the critical point. How is the critical point? Sir, dy by dx is equal to 0. Or dy by dx is equal to does not exist.
Okay. By doing this. Okay, so here let's assume that x is equal to x1, x2, by doing this, all these xn will come as critical points. Second thing we will do here, find values.
At all the critical points. That means at f of x1, f of x2. All the critical points, you have to get values at all the critical points.
All the critical points. Get values. And you have to get values at the boundary points as well.
Get values at f of a, f of b as well. And where will you get? Boundary points per B of values in a column. Okay? Now, the greatest value among all of these, the greatest value among all, what will you call it?
Global maximum. Global maximum. Okay? The smallest value among all of these, the greatest value among all of these, Least value among all.
What will we call this? Global minima. Okay, crystal clear to all students. So, the basic thing of Kulmila is that whenever you want to find global maxima, global minima, always first of all, do the critical point.
After finding the critical point, find the value of the function at all critical points. Find the value of the function at the boundary point. The biggest value among all these is called global maxima.
The smallest value among all these is called global minima. This is the whole concept. Okay.
I think all my students must have understood the story. Let's start the question. Okay. Let's take the maximum value of the function. We have a function here.
This one. In the interval, close interval 1 to 3. So, how will we get it out? First of all, given f of x. This is 2x cube minus 24x plus 107. So, f dash x will be 2, 3, 6, x square. minus 24, right.
So, for critical point f dash x equal to 0, so 6 x square minus 24 is equal to 0, x square is equal to 4, x value is plus minus 2, right. Now we have to find the maximum value, so where will we find the value of the function? At the critical point, at f2 Remember one thing, x is equal to plus minus 2 We write it like this, x is equal to plus 2 x is equal to minus 2. What is the domain here? The domain is given, close interval 1 to 3. So, you see, is it 1 to 3?
Absolutely. Is it between 1 to 3? No.
That means, x is equal to minus 2, critical point will not be there. Okay. So, x is equal to minus 2. is not a critical point.
Clear? Whenever you find a critical point, if there is a closed interval, you must check whether the critical point is within that interval or not. If it is within, then you will take it.
If it is not within, then you will not take the critical point. Clear? Because there is no function there. So, from where will the critical point be? So, where will we find the value?
Now, Now, we will find the value of f of 2 per function. and at the boundary point F of 1 and F of 3 Now in these where the value is the biggest that will be the maximum value and in these where the value is the smallest that will be the minimum value. So at F2 it will be 2x2 cube minus 24x2 plus 107 Ok.
So simply do it how much will it be? This will be 16 minus 48 plus 107 How much will it be? 107 and 16, how much is it? Reduce 48 from 123. Reduce 8 from 13, 5, 7 from 11, 4, 7. This is 75, F2 is 75. Similarly, take out F1.
So, how much will F1 be? 2 minus 24 plus 107. So... 107 out of 22, subtract 2 from 7, 5. This will be 85. Right. Similarly, calculate value on F3. On F3.
So, 2 into 27 minus 24 into 3 plus 107. So, how much will this be? 27 divided by 54. Add 54 to 107. Add 54 to 107. 7 x 4 is 11, 161. From this, subtract 72. 11 divided by 2 is 9. 7 divided by 15 is 8. Right? 89 will be there. Did I do it right?
89 will be there. So here, where is the biggest value coming? It's coming at 89. So 89 will be the global maximum value. Okay?
So 89 should be the global maximum value. I think all the students are clear. I'll give you a small tip.
If you find the maximum value somewhere, like you're saying maximum value, maximum value means find the biggest value. And if you're given a close interval, whenever you're given a close interval, always find the value at the boundary point. Because, maximum and minimum value can be obtained even at the boundary point. Is it clear? So, this is the thing.
You have to always get maximum and minimum value at the boundary point. Let's do one more question. Come here in this question. Here, an interval is given, from minus 2 to 2. We have to tell the minimum value of this function. So, here sir fx is equal to what we have given, fx is equal to x cube minus 3x plus 4. So, what we will do?
For critical point f dash x equal to 0, that means 3x square minus 3 is equal to 0, this will be x square minus 1 is equal to 0. So, this is x is equal to 1, x is equal to minus 1. Now, listen carefully, my children. Is x is equal to 1 lying between minus 2 and 2? Is x is equal to minus 1 lying between minus 2 and 2?
Absolutely, it is lying. That means, where will we find the value of function? We will find the value of function at f of 1 and f of minus 1. Now, there is a problem here. Sir, at what boundary point will we find the value of function?
Pause the video and tell me. Pause the video here and tell me in the comment section quickly, at what boundary point will we find the value of function? I think you must have written it down.
We will not remove it because there is an open bracket on minus 2 and 2. On minus 2 and 2, there is an open bracket. Open bracket means that these two points will not count in this interval. Means we will not find the value at the boundary point, we will find it at the critical point only. Understand this concept properly. So what is the value of f1?
1-3 plus 4, so what is it? 3-2, f-1. If we subtract 7, we get 6. Okay, clear?
What do we have to tell? We have to tell the minimum value. So, the smallest value of these two, minimum value. Where did the minimum value come?
It came on 2. So, 2 should be the correct answer. Okay, brother. Is it clear to all the children?
Yes or no? Come quickly and see. I think all my children have crystal clear.
Okay, Pattar? Any doubt? No, sir? Absolutely not. Let me tell you some important techniques and tricks.
If you ever get an objective question, then always keep it in mind. The first one is, if let's assume that A sin x plus minus b cos x is written somewhere, its biggest value is root a square plus b square. Its smallest value is minus root a square plus b square.
Okay. Remember this for the rest of the day. Second, if we assume that f is equal to sin x into cos x, then the method is this.
You multiply it from 1 to 2 sin x into cos x. So, this becomes sin 2x. Now, what will you put? Sir, from where sin 2x will lie? minus 1 by 2. So, where will 1 by sin 2x lie?
From minus 1 by 2 to 1 by 2. This is the second concept. Third, if parabola is given somewhere, Ax square plus Bx, I didn't want to tell you. If A is greater than 0, then parabola upward, then its smallest value will be minus d by 4. This will be its smallest minimum value.
And if parabola A is less than 0, then parabola will be downward. So its largest value, This will be minus d by 4. Okay. So, remember these two things.
Parabola upward is the smallest value. Parabola downward is the largest value. So, although you don't have to use them much. Okay. Use them in objective questions.
So, the first question, the maximum value is of sin x plus cos x. So, what will happen? So, 1 square plus 1 square is root 2 minus root 2. So, root 2 will be the maximum value.
So, keep this in mind. See here, its value is maximum value, so remember that it is 1 by 2. So, its maximum value is 1 by 2. I think all my students have crystal clear. Here, what is stationary point?
Stationary point is basically a point where dy by dx is equal to 0. Keep this in mind, this is called stationary point. If you say no to telling stationary point, then always do dy by dx is equal to 0. At the last time, if you understand the session, please comment below the video. If you want the PDF of this class, download the app of ADDA247.
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Koi bhi aur doubt. Hopefully, you have understood my children. You have made notes in this way.
Download the notes from telegram. Or email me from the community. You can download from anywhere. Make good notes.
Rest, there will be one-shot session. There will be practice session of one-shot. Be active.
Thank you so much for watching this session. God bless you all. Bye-bye.
Take care. Do tell in the comment section how was the session. I hope you have finished the first book. Now, we will start the second book i.e. Integral.
Okay, so I have done all the chapters of book 1, from chapter 1 to chapter 6, I have done all the detail one shots, everything. Rest will be substituted on this, which is the session of question practice. Then both will be combined and a good session will be made. Okay, so thank you so much for watching this session. God bless you all, all my children.
Bye bye, take care, all the best.