in this video we are going to look at the topic of optimization and calculus curves so this is quite a difficult topic to grasp and it is asked in a lot of the more difficult IV maths exam questions so we really want to get an understanding of what optimization means and how it relates to curves and their derivatives and their second derivatives so I have a function here I have my curve this is going to be an FX curve and I've just putted some I put some dots referring to some coordinates on our curve so we have point a here we know here X would be negative 2 here X would be 0 that's at the origin oh I've just put some point here at B and we also have 8 and then we have C so well what I want us to understand is what does f of I equal and this means what is the Y value at Point a so at Point a the Y value because this is our FX function will be 0 because on our Y axis it's not up here it's not down here it's actually when y equals 0 so f at this point here will be equal to 0 secondly I want us to think about what does f dash of I equal now f dash of a means what is the slope or what is the gradient at Point a so f of I meant the Y value F dash of a means what is the slope and at Point a well we don't know the exact answer of the slope but we do know that it's part of a line here that seems to be sloping up so the first derivative at a will be some positive answer so f dash of value will be positive and it's very different to what f of a is f of a is the Y value and dash of a is the slope and that'll be positive and then finally what is f double dash of a the second derivative so we have the second derivative the second derivative relates to the concavity so where our point is a is that part of a negative or positive or a zero concavity well it's part of this curve here and if it's curving down it's a negative concavity so we're going to get some negative value for the concavity at a so if we can understand what the difference between F F Dash and F Double Dash is you can really you can really understand these problem-solving questions okay let's go through the other points a little bit more quickly so F of negative 2 this means when X is negative 2 what's the Y value well are we positive it'll be up here so this will be a positive value now let's do F dash of negative 2 this means what is the slope when X is negative 2 or when X is negative 2 the slope will be 0 it's a maximum so this will be 0 and what is f double dash of negative 2 well at this point here it is part of the negative concavity so it will still be a negative value after a few of these of hopefully you can see the pattern or understand what these mean f of 0 so when x is 0 at the origin the Y value will also be 0 ok f dash of 0 this is going to be the slope at the origin and it's going to be sloping down so this will be negative and the double derivative which is concavity at 0 well this will still be part of the negative concavity section so we'll have a concavity which is negative okay 3 more points f of B so f of B means when X is at point B which will be here what is the Y value well this will be negative okay what is f dash of B well at point B here the slope is down so we'll have a negative slope and the double derivative this is an interesting one this one the double derivative is the concavity is it part of a the negative concavity or as a part of the positive concavity or looks to be right in the middle and if we have no concavity it's a point of inflection so this is a point of inflection and that's when the double derivative is equal to zero okay two more so we have f of 8 what does f of 8 mean so when we go to when X is 8 down here the Y value will be negative now the slope F dash of 8 what is the slope here well it's a turning point it's a minimum so the slope will be equal to 0 and f double dash will be our concavity and this point here is part of the happy face so it's going to have a positive concavity and the last one at Point C F of C the Y value right here will be 0 now F dash of C this is going to be part of a the point will be sloping up so be positive and the concavity at C will also be positive at it's part of our happy face here so if a double dash of C will be equal to some positive value okay so hopefully once we've got at all of that we now understand the difference between FX f dash x and f double dash x this just being our original function this being the first derivative this being the second derivative it's really important to understand that at any point X FX means the Y value F dash X means the value of the gradient or the slope and F Double Dash X is the value of the call so if we understand this we might be able to draw the first derivative or the second derivative if we are given the first derivative the original function and we can answer these tricky problem solving questions where they might give you a derivative curve okay so that's that's what the ow curves will look like with regards to this this word here optimization if a function has been optimized it means they need to be a maximum or a minimum so in this function here the optimum points the maximum and the minimum are going to be these points here the max and the min now you may think well the min is actually lower than this it's down here if the function continues and the max is actually going to going to keep going higher that's why you often hear these referred to as the local maximum and the local minimum because in a exam question they might restrict the domain to something between here and here where this is actually the minimum and this actually is the maximum but more likely you're going to get a real-world problem where you might get a surface area or a volume or or an area of a rectangle and what these questions ask for is what's going to be the the area or the width or the length of our shape when the and whatever it is will be will be optimized or be a maximum or a minimum so if you do come across these questions where you see the word a maximum or a minimum you need to follow these key steps so with optimization you need to find the first derivative find F dash X and when something is being optimized for example here at the min or the max notice that the slope is zero it's where a turning point occurs so the first derivative we now know refers to the slope so if you ever want to maximize or minimize something get an equation derive it find the derivative and make this equal to zero and if you do that and solve for where the X points are or in this case x equals negative 2 and x equals 8 you're finding where the function has been optimised a maximum or a minimum now to check if it's a maximum or a minimum there are a few different ways so you can find your your ways in your textbook or from a teacher but the most common way is to then use the x value that you found to be an optimum optimum place either a max or a min and if you sub that into the double derivative and it comes out as a positive answer it means that you got the minimum because the minimum is an optimum spot that's part of a positive concavity so if the if the double derivative is positive at that optimum spot it's a minimum or if the double derivative was a was a negative you had the maximum so that's the it's the most common way to find whether you have maximized or minimized the function okay so these questions they do take practice my goal for this video was for you to understand the difference between FF dash and f double dash if you want to optimize anything you need to get an equation derive it and make it equal to zero and that finds the maximum and minimum spots and then you can use the double derivative to test whether it was a minimum or a maximum okay I encourage you to practice a few questions so good luck