[MUSIC] Definition of a critical point of a function. Let f(x) be a C1 function, Defined on U, Where U is an, Open set. We say that x* is a critical point Of f, if the gradient of f at x* is 0. Well, clearly x* belongs to U here. We have considered three cases of three functions. We dealt with z = x2 + y2, z = -x2- y2 and z x squared- y squared. In all these cases, the origin, Is a critical point, Because while we find derivatives of all these functions we're going to expect through x and y these derivatives turns 0 at the origin. But we know that these three points are different because in the first case when we deal with z which is the sum of the squares the origin was the point of street local minimum. For this function 0 point was the point of the street local maximum. And, in the third case this point, the origin, wasn't an extreme at all. So when we look at the surface represented by this function as the graph of this function. We saw that it looked like a saddle and that means that we can find a new name for such a critical point. So for this function z equals x squared minus y squared, Is a saddle point. So we need to find the sufficient condition which will enable us to distinguish these critical points to classify them. And let's recall how it is done in the case of a single variable function. When we do with the function of one variable only, our critical point x* belonging to some interval I is a function where the first derivative turns 0. If you refer to the Taylor's expansion formula, let y star be the value or the function at x*. According to the Taylor's Formula, y minus y* is roughly f prime taken at x*, times x minus x*, plus one-half f double prime Times x minus x* squared. Or we can write this as the first differential the function taken at x* plus one half. Second differential of the function taken also to x*. But since x* is the critical point, and this first differential turns 0, so then the classification of the critical point depends on the sign or the second order differential. And the same true for the many variable case. The only thing is that we need to use Taylor's Formula for the function of many variables, where we'll be considering functions of only two variables So we need to use Taylor's expansion formula and we need to introduce second order differential such a function. But let me remind you first about the total differential. So given a function z which is a function of x and y, we call total differential Expression df, when we evaluate this expression at a given point such that we have df over dx taken at this point Times dx plus df over dy at the same point multiplied by dy We'll introduce this time second-order differential. By the formula. It is also calculated, at (x*, y*). And this is a combination of second order derivatives taken at this point, multiplied by the corresponding differentials of the independent variable. So we multiply by dx squared, break it somewhat. Use here plus, we continue here double mixed derivative, By the product of differentials of x and y, plus Here, we refer to the young theorem which makes two mixed derivatives equal each other under condition we here supposed of course. Otherwise, I won't be able to find these derivatives that f of x, y is twice continuous differentiable. Now Taylor's formula. Looks very similar for the previous formula written in a single variable case. So if z* is fx* y*, then the difference is z minus z* roughly equals df. So you can add (x*, y*) + 1/2 second order differential of this function, also taken at this point. Now, given a critical point, xy, x* y*, we know that the differential equals 0, equals both first order derivatives turns 0. So this is 0. Now the difference between the value of the function z. Taken in the neighborhood of the point x* y* and the value of the function itself at this point, entirely depends on the sign of the second order differential. If you look at the formula which represents a second order differential, you can tell that this is a quadratic form. Because these are numbers, in what variables, dx, dy. And quite soon, we'll be refreshing some facts from the theory or the quadratic forms. But at the moment, now let me consider a problem from economics. So, let's suppose we have a function, a production function. We dealt with it earlier. So this is a cubic route of the product of the factors of production, and let's state a profit maximization problem for a perfectly competitive firm, which is a price taker. When the profit should be maximize and we write it in the form of Talking about the first order conditions. We can find it easily by differentiation of the profit with respect to labor and capital. Then we get so, called first order conditions in the form of two equations. So, here we have, Should equal W and here By solving the system we can find the optimal values of labor and capital. The question is whether we are sure that this critical point is really a point of maximum of Pi. But then we'll have to check the sufficiency condition. [SOUND]