 So, why I am introducing this partial derivative? Because one can give a sufficient condition in terms of these derivatives to say that the point critical point is a point of local maxima local minima or it is a saddle point. So, let us write the theorem sufficient conditions for local maxima local minima. So, it says first of all consider the function f x y, x y in the domain D such that f x x, f x y, f y x and f y y all the four derivatives exist at some point a, b belonging to D. All the derivative exist define. So, let us look at the 2 by 2 determinant f x x at the point a, b, f x y at the point a, b, y x at the point a, b and f of y y at the point a. So, what is this? This is 2 by 2 determinant whose entries are the first row is the partial derivative, second order partial derivative f x x, second entry is f x y, f second row is f y x and f y y. So, this is called determinant at the point a, b. So, this determinant is called discriminant, it is called discriminant of f at the point a, b. You will get a number. Calculate this quantity, you will get a number. So, the test is in this number. So, let us write what is the test. So, on this delta a, b the discriminant at the point a, b is bigger than 0. This number which you calculate is bigger than 0 and the partial derivative with respect to x at a, b is also less than 0. The discriminant is bigger than 0, the partial derivative is less than 0. Then of course, we should have f x at a, b equal to 0 equal to f y at, they should be critical points. We are looking at critical points. So, these are discriminant. So, let the first derivative at a, b be 0. Look at the discriminant if it is positive and the second derivative f x x is less than 0, then f has local maximum at a, b. Then the point a, b is a point of local maximum. So, this is something like the second derivative test for functions of one variable. Looking at discriminant that is positive. If you look at the second derivative test for function of one variable, that was second derivative at that point should be less than 0 in one variable. That implied local maximum and same thing is here also. That is one conclusion. Second, discriminant at a, b is still bigger than 0, but f x x at a, b is other way around. So, other possibility is bigger than 0, then f has local minimum at the point a, b. Again, similar to the one variable. If the second derivative test, if the second derivative at the point a, b is bigger than 0, then the point was a local minimum. Same thing happening. So, this is when the discriminant at that point is bigger than 0. So, what is the other possibility? Discriminant at that point a, b is less than 0. In that case, f has a saddle point at a, b. Then it has saddle point at a, b. So, look at discriminant if it is bigger than 0 and look at the second order derivative with respect to x. If it is bigger than 0, local minimum less than 0, local maximum. If the discriminant is less than 0, then you can conclude it is a saddle point. What happens in case this is the, what is the possibility left? This is equal to 0. Then f may or may not have local maximum minimum. Means what? What does the statement mean? This means that the test, you cannot conclude anything. The function may have discriminant, which is equal to 0. Function may have a local maximum at that point. The function may have a local minimum at that point, may have a saddle point. Anything is possible or none. So, then you have to employ some other method of concluding something. It is like one variable if the second derivative is equal to 0, then you cannot conclude that the function has a local maximum or a local minimum. You may have to go to third derivative or some other method. So, these are only sufficient conditions to ensure that. So, probably let us look at some examples. So, we will have studied that. Probably I think because that will involve some computations. So, let us, whatever I have said is there. So, let us look at some examples. So, this is what saddle point sort of the saddle looks like is a better picture. So, you can have a look at. So, if you go along this, there is a maximum and along this, there is a minimum. So, that is saddle point. So, the definition once again saddle point p is belonging to the domain is a saddle point. If there is a point where the value is bigger than the value at that point and also there is a point where the value is smaller. So, let us look at the theorem and the consequences. This is the discriminant we defined. So, it is a 2 by 2 determinant f x x, f x y, f y x and f y y. So, they may not be 0 always, but sometimes you are lucky and they are. So, here is a test. If at a point, first derivatives are 0, that is necessary condition anyway for a point to be local maximum or minimum. We are analyzing the critical points. What we can say further, then if the discriminant is bigger than 0 and the second derivative x x x is less than 0, which will also imply actually that f y y, either of them can be looked at. Then the function has a local maximum and similarly, the discriminant is bigger than 0 and the second derivative with respect to x is bigger than 0, both are bigger than 0. Then this is a point of local minimum and if the discriminant is less than 0, then it is a saddle point. Equal to 0, you cannot conclude anything. The test fails, we can say. It is inclusive. So, let us look at some examples to illustrate this point. Let us look at this. f x y is equal to 4 x y minus x 4 minus y 4. So, the function is continuous everywhere. It is differentiable everywhere. No problem. It is a polynomial function of two variables. What are the partial derivatives? The partial derivative with respect to x. So, it will be 4 y minus 4 x cube. So, that is with respect to x and similarly, with respect to y, it is symmetric. So, you will get the partial derivatives. So, both equal to 0, you have to solve those two equations. So, when you want to solve these equations, this equal to 0, this equal to 0, simultaneously solve these equations. So, you will get points where the function can have local maximum and local minimum. So, this gives you three points, namely where possibly the function can have local maximum. So, how are these three points obtained by solving these two equations? So, we are not spending time on solving that. We will have to do that. Now, the point is at 0, 0, at 1, 1, at minus 1, minus 1, whether the function has local maximum, local minima or saddle point or you cannot say anything directly, you may have to look at something. So, let us try to see whether test is applicable or not. So, let us compute the discriminant. So, if you find out the second derivative, the minus 12 x cube square f x y is 4. So, that gives me, this is the second derivative. This is the discriminant at any point. Now, we have to compute it at the point separately for 0, 0, 1, 1 and minus 1, 1. So, let us look at the point 0, 0, when x is 0, y is 0. So, these two terms are 0. So, that gives you a negative term minus. So, that is negative minus 16. So, directly say 0, 0 is a saddle point, discriminant you compute. So, there is a computational application of 0. Similarly, compute at 1, 1. So, put x is equal to 1, y equal to 1. So, what happens to this thing? x is equal to 1, y equal to 1, that is 9 minus. So, that is bigger than 0. So, discriminant is bigger than 0. Now, we have to look at f x x at that point. So, you compute f x x at that point and that turns out to be minus 12. That is less than 0. So, what is the conclusion? Discriminant is bigger than 0. Second derivative f x x is less than 0. This is the point of local maximum. So, this is the point of local maximum. Similarly, for the other one, because it is product, so it will not change. The product will not change. Similarly, you can check that at the point minus 1, minus 1 also discriminant is bigger than 0 and f x x or f y y is also less than 0. So, both the points are points of local maximum. So, test applies to both of them. If you like, just look at the point 0, 0 as a point of local minimum. Local maximum? No, it was a saddle point according to the test. Let us just see whether we can avoid using the test. So, look at the line y equal to x, that is passing through 0, 0. We are looking at the point 0, 0. So, look at the line passing through 0, 0. What is f x x? That is this quantity, which is always bigger than 0 in a neighborhood of 0. Similarly, if you look at x minus x, that is negative. So, along the line y equal to x, the function remains positive. So, close to 0, 0, you can find points where the value of the function is positive. At 0, 0, the value is 0. So, along the line y equal to minus x close to 0, 0, you can find points where the value is negative. So, by definition itself, you can conclude that the point 0, 0 is a point of, is a saddle point for the function. You do not have to apply really the test, but test gives a nice straightforward application. But you can look at, so how do you analyze something? Test may fail. So, then how do you analyze whether it is a local, that is a saddle point or local. You may have to go to the definition straight away. See in a neighborhood, how does the function behave? So, let us look at, this is another simple example. For example, if you look at minus x 4, y 4, we know that what does the graph look like? x 4, y 4. So, it is always positive. It will be like a cup and the point 0, 0 will be a point of local. It is a negative thing. So, it is an inverted cup. So, point 0, 0 will be a point of absolute maximum. Not only, at every other point, the value is negative. But if you look at the discriminant, at that point 0, 0, what is the discriminant? First derivative will give you 4x cube and you compute that second derivative also and compute the discriminant that comes out to be 0. So, test fails. In this case, the test fails. Discriminant is 0 and the function has local maximum at the point 0, 0. So, if a test fails, it does not mean that anything is possible. Test fails means you cannot conclude anything while looking at the discriminant if it is 0. So, this is an example of a function where the test fails, but the function has a local maximum at the point 0, 0. And if I invert, that means if I take x 4, y 4 without the negative sign, then again the discriminant will still be 0. But the function will have a local minimum at that point. So, test may fail, but anything is possible. So, that is what this example is showing. Look at this example. So, in this example again, if you compute discriminant, that will turn out to be equal to 0. So, as such, test does not help you to do anything. It does not help you to compute or conclude anything. But for this, you can try to maneuver. For example, if I look at this line, say y cube is here. If I look at the line x is equal to minus y, what will happen? I can analyze this function along a curve passing through 0, 0. So, for example, I can look at the line where y is equal to minus x along that line. So, if you look at that, then what does this function look like? Is it negative? y is equal to minus. So, or look at, say, look at the value of the function x 1, t, y 1, t along the curve t minus t. Along that, the function takes the values negative. Along with some other curve, passing through it takes the value positive, y is equal to x. So, that means along some curve, it is the minimum. Along some other curve, it is the maximum. That is good enough to say that this is the point of saddle point. So, this is by analyzing the behavior of the function at points close to the point where you are looking at along some curves, but the curve should pass through that point. That is important. You should not look at arbitrary curve because you want to look at every neighborhood of that point, should have a point where the value is bigger and some other point where the value is smaller. So, you should have a curve passing through that. So, this is the way you analyze that 0, 0 is the saddle point.