 All these properties, there are points where the nature changes, they are called points of inflection. For example, if you look at y equal to the function x cube, how does the graph of the function x cube look like? It goes from the down, it is negative, x cube is negative on the negative side at 0 and then it starts increasing, but the nature rate it is different. It is concave down, it is concave up there and at 0 there is a tangent which is horizontal, because derivative of f of x cube is x square at x equal to 0 slope is horizontal. So, at 0 there is a tangent. So, it goes smoothly and cup, but the nature of the function on the left and on the right are totally opposite to each other, concave down to concave up. So, you say the point x is equal to 0 is a point of inflection. So, you can define in general a point of inflection to be a point where the function changes its nature from concave up to concave down or vice versa. All these are properties of the function. It helps you to draw a picture of the function. How does the function look like? So, there are examples. You can just skip those examples. The important theorems are the following. This is important that if a function is convex and you are given 3 points a, b and c. What is the first one? f b minus f of a that is the slope of the chord joining with b and a. c and a that is the chord joining c and a and this is the chord joining c with b. So, these are relation between the slopes of these 3 chords. So, let me probably draw a picture that you can see easily if this is the function and this is a and this is b and this is c. I hope that is the way it is taken a, b. So, look at this chord and look at this chord. What is the relation between these 2 slopes? Theoretically, you can just look at the slopes. Slope of a to c is more than the slope of a to b and look at this one now. At this point, this slope is and this slope, what is the relation? That is less. So, that is what this is saying that the slope between the 3 chords is as follows. One can prove it mathematically, we will not do it. As a consequence of this, one proves beautiful theorems about convex functions. Let me, it says at every point you will have a left derivative, you will have a right derivative and so on. So, there are lot of these results. So, just go through them once, read them once and forget them because I will not be asking them in the exam. But you should understand what this implies. For example, it says that plus, this is the derivative from the right side f dash plus, plus is on the right side and the left side derivative exists at every point. It is a beautiful thing. And the right derivative is always less than the left derivative. You can, because of that slopes actually, basically taking limits of them and eventually all of them lead. So, I say accept it. So, here is something interesting. A convex function is differentiable everywhere except at countably many points. It need not be increasing. It could be like in some parts increasing, some parts decreasing, but still it has that properties. It has to be, and of course, differentiable implies and it is continuous everywhere. So, very nice properties of convex functions and you may come across these kind of things in your courses in probability and statistics. So, that is why exposure is a good idea. One can state results, sufficient conditions for a function like derivative gives you the nature of the function increasing or decreasing. Similarly, nature of the second derivative gives you whether it is a convex or concave. So, essentially this is the what, if f is concave upward if and only if derivative is f dash. So, you look at the slope of x square at 0, negative becomes 0 and then start increasing. So, that is f dash is increasing. And if second derivative is here, when is second derivative? If a function has second derivative and f dash is a function which is increasing, what is the nature of the second derivative? f dash is increasing. So, what can you say about the second derivative? Just now we said, increasing derivative exists, derivative should be bigger than or equal to 0. So, say second derivative should be bigger than or equal to 0. So, that is the second derivative test for convex T and concave T. Function may not have second derivative. If it has, then that should be the property. So, that is the first derivative test and this is the second derivative test. f concave upward if and only if second derivative is bigger than or equal to 0. So, all this is because we know the function is differentiable once or twice and we know the property of the derivative or the second derivative. It gives you back the knowledge about the function. That is the important thing. So, that is convex T and concave T. Let us start looking at the, we have time. So, let us start looking at. So, in basically in one variable, what I have tried to give you a field for what is derivative? Why we consider derivative, a function of one variable? The important property of derivative is that it gives you some kind of a smoothness of the function that is saying at every point, you can draw a tangent and the derivative gives you the slope of the tangent. Derivative does not give you the tangent. Quite often, it is misunderstood and misstated that saying function is differentiable. Derivative is the tangent. Derivative is not the tangent. Derivative is the slope of the tangent at that point. Tangent is a line which you have to draw. Once you know the slope, you know the point, you know the tangent line. So, and the important algebra of differentiable functions that we just went through, assume that you have all gone through this kind of theorem before. Derivative of addition, subtraction, product rule, quotient rule, chain rule and so on. Important thing of derivative function being differentiable also is that you can approximate the values nearby with the values at that point and the derivative. That is the important way of saying what is differentiability. And then we saw that if you have, if you know what is the derivative, you know some property of it, then that gives you back some properties of the function and the main theorem is called Lagrange's mean value theorem which gives you the applications. So, let us now look at a function of several variables. F is a function defined on interval. So, let us look at domain D. We will do it in R 2. Same analysis works in R 3 and so on and R n. We have already analyzed continuity of functions of several variables in the previous lectures. And we try to look at, given this function, we, if we fix x naught and y, right. So, y going to R, you fix for fixed x naught and x going to f of x, y naught, fix y naught. So, they were the coordinate functions, right. So, a function of two variables, once you fix all but one coordinate that gives you the coordinate functions. And we saw that continuity of a function is much more important than saying each coordinate function is continuous. So, now let us look at differentiability. We want to come to a notion of differentiability of functions of two variables, right. And we would like to say that differentiability should enable us to give something what we have done in one variable. And what are these things? Notion of tangent I should be able to define in one variable, right. And I should be able to say it is smooth. And smoothness was saying that tangent is possible, right. Now, what is the meaning of saying corresponding thing if you want to translate for functions of two variables, right? In one variable, smoothness was we are able to draw a tangent at every point to the graph of the function. What should be the corresponding thing for functions of two variables? So, let us look at the graph of a function of two variables. What is the graph of a function of two variables? So, f is a function. So, for this function, what is the graph of the function? Before we want to draw something like tangent, we should know what is the graph? What is the graph of a function of two variables? Or what is the graph of a function of one variable? All the points in the plane x comma f of x, right. Every point in the graph is x comma f of x. So, here this is x and this is h of x. So, what is this point? Coordinate of this point is x comma h of x. So, all these points are x comma h of x, right. So, here what is the domain? Domain is x y, two points and what is the value f of x y. So, what is this point? If I know this thing, I know the function for every point x y belonging to the domain. If I know this triple, I know the function. At x y, the value is f of x y. So, this is the point in R 3. This is the point in R 3, x y. So, what does it look like? y and here is a z coordinate. So, here is the domain d. You take a point x y. We have done it when we defined what is the function of two variables. What is f? It takes a point here and gives you the value f of x y. So, that is at height. So, at every point in the domain, you look at those points in space. So, that gives you something like a surface in R 3. So, graph of a function of two variables is a surface in R 3. For two variables, it is a curve in R 2. Now, it becomes a surface in R 3. So, what I want to do is for this surface in R 3. So, this is my surface. At any point here, I want to draw something like a tangent. I want to say this is a smooth surface. So, when you say physically, what would you say this kind of thing is smooth? When would you say this is something smooth? I am not asking mathematical. You are saying when we say this plane is very smooth. If there is a stone lying here, then you cannot just glide over that thing. You have to sort of do the obstacle. Look at a hill with stones inside it. So, there is not there are obstructions. You cannot glide kind of a thing. So, saying something is smooth means you should be able to draw a plane which should touch the surface only at one point. That you would call as the corresponding thing of tangent line for one variable function. The graph of a function of two variable at every point, I should be able to draw a plane which touches the graph only at one point, that point. So, how do I capture that thing problem? Now, a plane is known. See, there a tangent line was known once I know the point and the slope. Now, if I want to find out the equation of a plane geometrically, I know the point through which it is going to pass. This is the point. I know that at this point. I know the plane if I know a normal to the plane. I know the equation of the plane. If I know that at this point, this is the normal. So, given a line, given a vector which you would like to call as the normal, there is only one plane. There is only one object in the plane, object in R 3 which you can call as a plane to which this will be the normal and that you would like to call as a tangent plane. So, here the problem would be first to define what is a normal and the plane which is perpendicular to that normal will be called as a tangent plane. That is how we are going to go about functions of two variables. So, to draw that normal, what we do is the following. Some elementary things. First, consider our function is f d in R 2 to R. We will assume d is nice like an interval. It is a ball on which the function is defined. So, consider fix, say y naught, belong to the plane. So, consider y naught belonging to R, such that x y naught belongs to d. And look at x y naught going to f of the coordinate function. Look at the coordinate function. Let me draw a picture here on the side. Here is the domain, z, x. So, this is the point we are fixing. This is the point we are looking at. So, we have fix y naught. I am varying x. So, where I am moving? So, in the domain, I am shifting x. So, I am moving along this line, which does not look like parallel to x. So, let me draw it slightly better. So, this itself looks like this line. So, what is this line? Green one. These are the points x for a point x, x comma y naught. So, that is the point here. So, this point is x comma y naught. At this point, look at the value of the function f of x comma y naught. So, what will that give me? As I vary x, so that will give me a function of one variable. So, that will give me a curve. So, at every point, this is the point of this. So, this is x naught y naught, f at x naught y naught. So, at every point, now I can ask the question for the coordinate curve. Does it have a tangent at that point? That means, I am asking, does it have a tangent at this point? If it has, then we will say the function has partial derivative at the point x naught y naught in the direction of x. So, look at d by dx of f x y naught at the point x is equal to x naught. If it exists, then you will write it as f dash x naught. I should not write f dash because it is only with respect to x naught y naught partial derivative of f with respect to… So, instead of writing d by dx, you start writing del delta. So, this is, what is this quantity? This is the derivative of the coordinate function y naught fixed, x is varying. So, x is varying. Similarly, you can also ask for whether the other coordinate function is differentiable or not at that point. So, you will have partial derivative of f x naught y naught with respect to the variable y. They may not exist, like function of one variable y, derivative exists. But if it exists, so these are called the partial derivatives of f at the point x naught at x naught y naught with respect to the two variables, the function of two variables. So, we are looking at a function of three variables. You will have partial derivative with respect to the third variable also z. So, the function… So, let us look at some probably examples before we go further. I think we looked at that example for continuity of function f of x y. So, let me write as x plus y divided by x minus y. If x y naught equal to 0, 0 and 0, if x is equal to 0 equal to y… Oh, sorry, not… It is x minus y. So, I should write a bit more carefully, because the domain should not have x is equal to… If x is not equal to y and 0, otherwise, let us write… So, domain of the function is r 2 minus the line y equal to x. And we saw this function f is not continuous at 0, 0. We saw that this function is not continuous at 0, 0. Let us try to look at the partial derivative of this function at the point 0, 0 with respect to x, whether it exists or not. So, what is this partial derivative? It is a limit h going to 0, f of with respect to x. So, h comma 0 minus f at 0, 0 divided by x. So, what is this equal to? Limit h going to 0. What is x 0? What is the value of the function? That is equal to h plus 0 divided by h minus 0 minus f at 0, 0, 0 divided by h. So, what is this? It is a limit h going to 0. This cancels out 1 over h. That does not exist. h goes to 0. We have to look at limit h going to 0. So, the partial derivative of this function does not exist with respect to x. Similarly, you can analyze with respect to y. So, this function is neither continuous nor the partial derivative of this function exists. Let us just look at one more example before we conclude or stop the lecture today. Let us look at x y divided by x square plus 1, say. Let me write x square plus y square. So, then I have to modify x y not equal to 0, 0, equal to 0 otherwise. Now, look at the function partial derivative of f with respect to x at 0, 0. I want to find that. So, I should look at the value of the function when y is x is varying. So, y is 0. So, when y is 0, this function is 0, constant function. So, equal to 0, that is same as the partial derivative with respect to y at 0, 0. When I fix one of the variables x at 0 or y at 0, the function is constant function 0. So, derivative exists. Is it continuous? Is f continuous at 0, 0? So, one of the ways of looking at it is look at a along a curve. How does the limit look? So, look at y equal to m x. So, what will the function look like around the line y equal to m of x? Line passing through origin, the path should pass through the origin. We saw last time. So, that will be equal to m x square divided by m x square plus m square x square x square will cancel. So, limit is m square m divided by 1 plus m square. So, that depends upon the slope of the line. So, along each line the slope is different. So, the limit is different. So, the function is not continuous at 0, 0, not continuous. So, existence of partial derivatives at a point need not imply the function is continuous. So, analyzing differentiability of one variable at a time is not good enough to say that the function is differentiable. We need something more than saying both the partial derivative. Like continuity also, continuity of function in each variable need not imply continuity jointly. So, we will see what is that required to say that the function is differentiable as a function of two variables. So, we will stop