 Let us look at some more properties of. So, let us say, so we will assume say f is differentiable. I am just revising most of the things, which you already know, may be slightly giving a different perspective of the things. Then one would like to know, we I also already mentioned that if a function is monotone, then we know it is continuous except at countably many points. In fact, I pointed out that there is a deep theorem saying that if f is monotone, then it is also differentiable except at some number of points. That means, most of the time the graph should be a smooth graph except at some points, which are which have probabilities 0, forget about that statement anyway. So, f is differentiable at the point see. We would like to know, if you know something more about the function, can you give me back some more properties of the function? For example, suppose I know the function is differentiable that gives me continuity. But I suppose I know something more about the notion of derivative, that how does the derivative of the function look like? Can you give me back some information about the function? So, for example, let us try to understand it. Let us take the, so this supposing some function and here is the graph of the function for some portion at the point C. Graph sort of is going up and then starts going down at this point f of C, what we call as a local maximum. So, what is the meaning of local maximum? It is a maximum for the function, but not everywhere, the graph could go up somewhere. But in some interval C minus delta to C plus delta, I can say there is a interval around the point C such that there is a neighborhood of the point C, so that the value of the function at the point C is the largest. Then we say the point C is a point of local maximum, we all have gone through this. Now, if the point of C is a point of local maximum and there is a notion of derivative available at that point, that means I can draw a tangent to the function at that point, then what should this look like? What should the tangent at that point look like? At this point, there is a tangent possible. So, it looks like the tangent should be horizontal one. So, geometrically, we are guessing. If f is differentiable at a point C, at a point C, f has local maximum at C, then we are saying that the derivative at that point should be equal to, so this is a geometric observation. You can prove it very easily. So, let us just give you a proof, which you might have already done in your courses earlier. So, I want to calculate the derivative. I already given that it is differentiable. So, what I want to calculate? f of C plus h minus f of C divided by h limit of this as h goes to 0. I want to calculate the limit of this as h goes to 0 and I want to show it is equal to 0. So, let us analyze when h is positive. If h is positive, what happens to the denominator? h is positive. So, a denominator is positive. Function is increasing. So, what happens to the numerator? Function is at this point, it is a local maximum. That means if I take any point on the left side here, then the value at this point will be less than the value on the point C. So, this numerator will be negative, because f of C is the largest value in the neighborhood. So, the difference will always be negative. Do you want a denominator if h is positive is positive? So, ratio is negative if h is positive and the limit of that, we know it exists. If you take it limit from the right, I am taking the limit from the right now, h positive. So, that will be less than or equal to 0, because the ratio is always negative. So, limit must be negative. It exists. Is it okay for everybody? Now let us look at the same limit. So, here I am taking h bigger than or equal to 0 of this quantity. Let us take the limit of this quantity from the left side. h is negative now, but the numerator is still positive, because the function is local maximum at the point C. What is the numerator? Still negative, denominator negative. So, ratio is always bigger than or equal to 0. So, limit should be bigger than or equal to 0. I am just looking at the function for which you are taking the limit. If h is positive, the function values are all negative less than or equal to 0. So, limit must be less than or equal to 0. If I look at the left limit, the function is always positive. So, limit must be positive. So, that implies f dash of C must be... So, the left derivative, right derivative, both are same, because the function is differentiable. So, it must be equal to 0. So, that is the reason. So, this geometric picture, we can prove it very easily by looking at this thing. So, this gives a very important theorem, namely what is called Roller's theorem. And what is that? So, I think most of you must have gone through it and remember it. So, that says, if f is a function on a interval a, b to r, closed interval a, b to r, f continuous, f differentiable, at least on the open interval a, b. And the third property says that f of a should be equal to f of b. Then, that implies there is a point C, right, such that, right, differentiable. So, derivative at the point C must be equal to 0. Geometrically, how one guessed this result is as follows. Here is the point a, here is the point b. So, f of a, here is f of a and that is equal to f of b. So, that is f of b here. So, that is equal to f of b. So, it is a continuous function. So, let us draw the graph. It starts here. It has to end here, right. So, what should happen? You can start going up or down or something. So, let us say it goes like this and comes like this, right. I should not lift. That is 1, because it is continuous, okay. Now, apparently the graph says there are many points where the slope of the tangent is 0, right. In this, it says here, here and here or even here. What are these points? These are the points where there is either a peak or a trough, right. So, it looks like these are the points where the function has local maxima or minima. So, if I can say if f is continuous, right, I can ensure that in the interval a to b, there will be at least one point of local maxima or local minima. Then I am through by the previous theorem and that is our theorem that if f is continuous on a closed bounded interval, then it attains its maxima and minima at some points in the interval a, b. So, f attains maxima, minima in the interval a, b. One possibility, maxima equal to minima equal to the end points, right. Then what is the graph of the function? Maxima is equal to minima equal to the value at the end points. There is a constant function. So, every place the derivative is equal to 0. So, no problem at all. Proof over. If not, that means there is at least one point in the interval a, b, right, where the function takes maxima or minima. That is inside the interval a, b. If it is a maxima or minima, automatically that point also is local because it is a global thing, right. It is the largest value of the function on the whole of a, b. So, locally also it is true. So, the previous theorem applies. So, basically it is a consequence of the fact that a continuous function or a closed bounded interval attains its maxima and minima and the previous thing that if the function is differentiable, then at local maxima minima derivative must be 0. These two combine together. And this theorem, you can easily extend this theorem by removing this condition, right. Look at the graph slightly tilted. That means what? f of a need not be equal to f of b. Then what does the graph look like? So, let us extend this theorem. So, let us say this is a, this is b and f of a to f of b. So, let us say it is value here and this is a value f of a and that is f of b and the graph is something continuous still. So, f on a, b continuous differentiable in a to b. We do not need end points. Differentiability is clear from the earlier because if inside the local maxima or minima, then derivative at that point is 0. So, f differentiable in a, b, we are not putting the condition that f of a is equal to f of b. We are omitting that. Then what should happen? So, let us look at the line joining this. Imagine this line to be moved up and down. Then at some point, there is a point where the tangent is parallel to this chord. So, there is a point c because in the Roller's theorem what is the slope of the chord? f b minus f of a, if f of b is equal to 0, so you get 0. Otherwise, it is a generalization of that. In general, it implies there is a point c belonging to a, b such that f dash of c is equal to f of b minus divided by mu. That is the slope of the chord. So, this is what is the famous called Lagrange's mean value theorem. And the proof also is straight forward. We just apply, try to apply Roller's theorem to a modified function. So, what is a modified function? f of a is not equal to f of b. But if I look at this end points, at that nothing values are not equal. But if I will subtract that, say look at the chord and look at this value and this value, this is the height h. So, this is at any point x. This is the value of the function f of x. This is the value of the chord. So, what happens to this height h as you move the point? If you move this point x towards a or b, what happens to h? At a it is 0, at b it is 0. So, that is the function I should be looking at. So, look at the function h. So, call h of x equal to, you can take l of, you can call this chord as l x. So, l x minus f of x. Consider that function. That function has the property that h of x is equal to h of a equal to h of b equal to 0. It is continuous because the function l x straight line that is continuous, that is differentiable. The difference is also differentiable and points, the values are equal. So, Rollins theorem apply. So, there is a point where the derivative is equal to 0. So, what is the derivative of h? It is the derivative of the chord that is f b minus f of a divided by b minus a. And what is derivative of f? f dash. So, f dash minus l dash is equal to 0 and that is precisely this consequence. So, that is how you prove Lagrange's mean value theorem from Rollins theorem. Or you can just think of, if you can visualize, rotate your axis a bit. So, that f of a equal to f of b. It should be true, but we are not going to that kind of argument. We are just looking at a straightforward way of saying it. So, that is called Lagrange's mean value theorem. So, this is one of the most important theorems, I would say of calculus. It says, if f is continuous, f continuous, differentiable on at least a to b implies f dash of 3 is equal to f of b minus f of a divided by b minus a for some c belonging to the open interval a b. Let us see what are the consequences of this, which we have all gone through. So, I will not go through all of them again. For example, suppose the function is such, now you see how the derivative is giving back you the properties of the function. f is differentiable on the interval a b. And assume, I know something more about the derivative, namely, derivative is 0 everywhere. Suppose the derivative is 0 everywhere. If derivative is 0, then what does f dash of c? That is 0. That means f of b must be equal to f of a. But why b and a? I can apply it to any two points in between x 1, x 2. Roller's theorem, Lagrange's mu values are applied to any two points x 1, x 2 inside a b. Then f of x 2 minus f of x 1 divided by x 2 minus x 1 must be equal to derivative at some point in between. That is 0 anyway. So, it says f of x 1 is equal to f of x 2. So, if f is differentiable in the interval a b, then it is constant. So, that is the consequence of this beautiful theorem. Very easy consequence. So, one f constant implies f differentiable. Sorry, f, what I am saying I should say, if derivative is 0, f dash of x equal to 0 for x 1 is equal to 0. So, every x belonging to a b implies f constant. In fact, I can say f is constant. See, I will get a point c in between. But you can apply it to what is the value at the point a? If it is 0 inside a b, it should be 0 at a because it is continuous. Function is continuous on a b. Lagrange's mu will be theorem. It says in the open interval a b, it is constant. But, it says that it should be 0 on the end points also. So, it is on the closed interval a b. So, this is one of the simplest kind of applications. There are more applications of this. What are the other applications of this? Now, you know that, for example, let me just, I will not go into that. We will just state those theorems for you to read. For example, look at derivative is 0. Suppose, derivative is bigger than or equal to 0. Let us analyze the second case. Derivative is bigger than or equal to 0. Then, what you can say about this ratio? Does it give you something? So, let me, so second that f dash of x bigger than or equal to 0 for every x belonging to a b. So, let us say this one. Then, what does it give me? By Lagrange's mu value theorem f dash of x for any two points x 1, x 2 in this interval. What you will get? There is a point x. So, let me just, I am just hurrying through. Let me not hurry through. So, let me write. So, for x 1 less than x 2 between a and b implies there exists a point x. So, either f dash of x is equal to f of x 2 minus f of x 1 divided by x 2 minus x 1. I am applying Lagrange's mu value theorem in the interval x 1 to x 2 which is inside the interval a b. And if this is bigger than or equal to 0, what does that give me? Whenever x 2 is bigger than x 1 denominator is positive. So, numerator should be positive. So, what should give you? If x 1 is less than x 2, then f of x 2 is bigger than f of x 1 function is monotonically increasing. So, how the nature of the derivative is giving you back bonus points about the function all because of Lagrange's mu value theorem? If the derivative is bigger than or equal to 0, function is increasing. Same proof less than or equal to 0 function is decreasing. So, that is the consequence of Lagrange's mu value theorem. So, it is telling you the nature of the function. So, now from here you can build up at a point c. I want to analyze whether there is a local maxima or minima for the function at that point c or not. Necessary condition derivative must be equal to 0. So, look at the points of the function where the function is differentiable. Derivative equal to 0. Analyze those points whether they are points of local maxima or minima and also the points where the function may not be differentiable, but still can have local maxima and minima. For example, mod x has local minimum at the point 0. It is not differentiable. So, that is not a sufficient condition. So, look at all the candidates namely where the function is not differentiable or function is differentiable and derivative equal to 0. Out of all these points some points may be local maximum, some points may be local minimum, some points may be none. So, how do you analyze what are the sufficient conditions? The sufficient conditions are if derivative is positive on the left side of that point, derivative exists on the left side in a neighborhood of that point on the left side. Derivative is positive. That means what function will be increasing on the left side or because of this theorem. Derivative is to maximum or decreasing. So, derivative is less than or equal to 0 on the right side. So, it will be decreasing. So, it will have a local maxima at that point. So, what are the points? f dash should be equal to 0, 1. On the left function need not be differentiable, but look at on the left side of that point and on the right side. On the left side if the derivative exists and derivative is bigger than or equal to 0, then the function will be increasing. On the right side if the derivative is less than or equal to 0, then the function will be decreasing. So, you get a sufficient condition f is c is a point. If function should be continuous of course, function is continuous and on the left the derivative is bigger than or equal to 0. On the right derivative is less than or equal to 0, then the function will have a local maximum and you can go on local minimum similarly. So, all of you have gone through these kind of theorems in your B S C courses, under the course. So, we will just take those theorems, we will not prove those. So, these are all consequences of for example, then you can go on to analyze what are called second derivative test. These are called first derivative test, you can go to second derivative test and so on. You can also analyze what are called convexity and concavity of functions. So, let me probably, I do not know how many of you have gone through, but I think it is a good idea to go through some of them. Let me just show you on the slide. So, maximum minimum that we have seen, what motivates one to define. So, algebra of derivatives, increasing, decreasing, we defined earlier also monotone. So, local maximum minimum definition, we define on the left side. In a locally it is a maximum or locally it is a minimum. So, here is a necessary condition for local maximum that if has a local maximum and is differentiable, then the derivative must be equal to 0. So, this is a necessary condition. Keep in mind, this is a necessary condition and how necessary conditions are used at points where the function is differentiable, but derivative is not 0, cannot be the points of local maximum minima or the function is not differentiable. That is a point of continuity. So, possible candidates for local maximum minima as a consequence of this are the points where the function either is not continuous or it is differentiable and derivative is equal to 0. So, that gives you a bag full of points called the critical points. You have to analyze which of them are local maxima or local minima. So, Roles theorem says continuity in the interval a to b. We said that this is the Roles theorem. So, the condition of end points value equal to removed. I am just revising again what I have said. So, applications of this consequences of derivative is 0, the function is constant. So, if the difference of, if two functions have the same derivative, then they may differ only by a constant as a consequence of that. So, this is about increasing and decreasing. If derivative is bigger than or equal to 0, the function is, you can prove other way around also. If the function is differentiable, f of x 2 minus f of x 1 divided by x 2 minus x 1 will be always, the ratio will always be positive. So, limit will be positive. So, other way around, if the function is monotonically increasing, then the derivative should be bigger than or equal to 0. So, this is a if and only if theorem. Similarly, decreasing strictly bigger than it is only one way. If f is strictly bigger than 0, then f is strictly increasing because x 2 minus x 1 will be strictly bigger than 0, equal to derivative. So, that is only one way. So, that caution one has to keep. So, continuity test for local maximum enema, as I said function is continuous at the point. On the left, I want the values to be less than or equal to. One way of saying that in terms of derivative is, derivative is bigger than or equal to 0. So, on the left it is increasing, on the right it is decreasing. So, that is one way of just increasing, decreasing straight away in the way, comparing the values. So, that is a continuity test. You can have the first derivative test also, we will do it later, may be. Here is something called concave upward and concave downward functions. So, what is the need for that? That is another property. So, you can have a function, say which is this function is monotonically increasing. This function is continuous, this function is smooth. But look at this function. So, this is f, this is g. g is also monotonically increasing or decreasing. It is continuous, it is smooth or you can look at this function. Look at this function, h, compare f and h. Both are monotonically increasing, both are continuous, both are differentiable, but there is a difference between the two. What is that difference? How do I capture that difference between the two? In some sense, the graph of f is bending away from axis. f of h is bending towards, that is all English. What is mathematics? So, mathematically it says, for this function f, if I take any two points and join, any two points and join, what happens to the chord that always stays above the graph of the function. In this, the chord will always stay below the graph of the function. And I can now make it mathematical at any point x. I know this chord, I know the value at this point, I know the value at the function. So, f of x should be, this is f of x and this is the value at this point is f at on this chord. So, call it l of x. So, the value at the chord should be bigger than or equal to value at the function at that point, then it is bending towards. So, you say it is concave down, if you like to call it, concave up. So, it is concave up, this is concave down. So, let us formally, so this is how you capture mathematically the properties. So, this is, look at f of x. If f of x is, what is this right hand side? This is the value of the function at the coordinate x, the point x hitting that line, the chord. This is the equation of the state line between x 1 and x 2. What is the slope of this line? f of x 2 minus f of x 1. So, this is the line joining the points x 1 with x 2. So, f of x is less than or equal to the value on the chord, on the line joining. So, you call it concave upward or convex functions. We will just keep the definitions will not, from examination of point of view, I will state some theorems which will not, because the proof are slightly complicated. Similarly, strictly if this inequality is strict, whether a constant function monotonically increasing, will be both convex and concave. So, if it is bigger, other way round, so you say concave down. So, for example, if you look at f of x square, the parabola. So, imagine your graph as a cup in which a spoon is lying, the spoon never touches the bottom. It only touches the rim. That is a parabola. You can think of a parabola. The chord joining any two points is always above the graph of the function. So, that is cup up. So, you call it concave up. So, concave up is y equal to x square. Other way round, you can take the graph of y equal to minus x square. So, that is other way round. The graph chord is always below. So, that is concave down.