 Hello, and welcome to this screencast on section 10.7, Optimization. As we saw in single variable calculus, the derivative is a useful tool for finding the local maxima and minima of functions, and that these ideas may often be useful in applied settings. In multivariable calculus, we will see that the same principles hold. We can extend the definitions of maximums and minimums to functions of two variables as follows. If f is a function of two variables, we say that f has a local maximum at a point x naught, y naught, provided that f evaluated at this point is greater than or equal to f evaluated at all other points near x naught, y naught. In this situation, we say that the output f of x naught, y naught is a local maximum value. Similarly, we say that f has a local minimum at a point x naught, y naught, provided that f evaluated at this point is less than or equal to f evaluated at all other points nearby. Here we say that the output f of x naught, y naught is a local minimum value. We say that f has an absolute maximum point at x naught, y naught, provided that f evaluated at this point is greater than or equal to f evaluated at all other points in the domain. In this situation, we say that the output f of x naught, y naught is a maximum value. Note that an absolute maximum point is an input that gives the largest possible value of f. And finally, we say that f has an absolute minimum point at x naught, y naught, provided that f evaluated at this point is less than or equal to f evaluated at all other points in the domain. In this situation, we say that the output f of x naught, y naught is a minimum value. Note that an absolute minimum point is an input that gives the smallest possible value of f. A bit more terminology that should be reviewed from single variable calculus, we use the term extremum point to refer to any point x naught, y naught at which f has a local maximum or minimum. In addition, the function value f of x naught, y naught is an extremum that is called an extremal value. Extremum points and single variable calculus always occurred at critical points. We have a similar definition for critical points of two variable functions. A critical point for a function f is a point where both partial derivatives equal zero or where one of the partial derivatives fails to exist. Critical points can help us determine the location of possible extrema, however, just like in single variable calculus, not every critical point leads to an extremum. Considered here, we have examples of three functions. On the left, a function with an absolute maximum. In the middle, one with an absolute minimum. And on the right, a function with what we call a saddle point, named as such because the surface looks like a saddle. All three of these functions have critical points at the origin, however, we see that the saddle point is neither a local minimum nor maximum because points nearby have values less than or greater than the value of the function at the origin. As in single variable calculus, we would like to have some sort of test to help us identify whether a critical point is a local maximum, local minimum, or neither. For this, we have the second derivative test for two variable functions. Notice that we have a critical point where both partial derivatives equal zero and let capital D be the quantity defined here using second order partial derivatives of F. We have several cases depending on the value of D and the value of the second order partial derivatives of F. First, we have if D is greater than zero and the second order partial derivative with respect to X is negative, then we can conclude that F has a local maximum at the point X not why not if D is greater than zero and the second order partial derivative with respect to Y is positive, then we can conclude that F has a local minimum at the point X not why not if D is negative, then we can conclude that F has a saddle point at X not why not. And lastly, if D is equal to zero, the test is inconclusive. The quantity capital D is called the discriminant of the function F. The second derivative test from the previous slide helps us classify critical points of a function, but it's not tell us if the function actually has an absolute maximum or minimum at each such point. In single variable calculus, the extreme value theorem guaranteed the existence of absolute maximums and minimums of functions. This theorem also applies more generally to functions of two variables. And it states that if a function F is continuous on a closed and bounded region R, then F has an absolute maximum and absolute minimum on R. Furthermore, these absolute extremes must occur at a critical point on the interior of R or at a boundary point of R. Just a note here on what we mean by closed and bounded in the extreme value theorem. This region here is closed because it contains the points on the boundary. That is, it contains points such that X squared plus Y squared equals one. On the other hand, this region is not closed because it does not contain the points on the boundary where X squared plus Y squared equals one. Lastly, when we say a function is bounded, we mean that it does not stretch to infinity in any direction. In section 10.7, you will get practice applying the extreme value theorem to find absolute extrema of functions.