 When we try to find an extreme value of f of x, we find the critical values, then determine which critical value corresponds to an extreme value. Well, that only works with functions of one variable. When we try to find an extreme value of a function of several variables, we should find the critical values, determine which critical value corresponds to an extreme value. Finding the critical values is not essentially different. Unfortunately, finding whether the critical value is an extreme value is. So here's a bad, well, actually a good, well, okay, an important example. Let's suppose we want to find the extreme values of a function f of x equals x squared plus y squared minus 3xy. So as in single variable calculus, we might find the critical values. So remember, these are where the derivative is zero or doesn't exist. And since f is a function of two variables, we use our partial derivatives and we find. Now remember, the reason we find the critical values is that if our derivative is non-zero, then our function is either increasing or decreasing through the value. And this is still true. Our derivatives are defined everywhere, but if either is non-zero at a point, then our function will not have an extreme value at that point. And so we want our derivatives to be zero, solving, and x equals zero, y equals zero is the only critical value. Now if this was single variable calculus, we might do something like apply the first or second derivative test. And this is one of the very few cases where that second derivative test isn't too terrible because we can find the second derivatives easily enough. And since both second derivatives are positive, the graph is concave up and the critical value is a minimum. At least this is what would happen in two dimensions, but we're actually in three. So remember, the partial derivatives come from viewing the intersection of the surface with a plane parallel to the x or y axis. So if we cut the surface with a plane parallel to the x axis, we see that the curve is concave up and our critical point corresponds to a minimum. But if we cut the curve with the plane y equals x, then the equation becomes, which means on the plane y equals x, the curve will be an inverted parabola. And so our critical value is neither a maximum nor a minimum. Because, looked at from some directions, it's a y-axis. So the curve is concave up and our critical point corresponds to a minimum. But if we cut the curve with the plane y equals x, then the equation becomes, which is a maximum value. But looked at from others, it's a minimum. We still want a name for it, and so we say that a critical value that corresponds to neither a maximum nor a minimum is called a saddle point. To determine whether a critical point corresponds to an extreme value, we need to examine the second partial derivatives, including the fixed partial. And we introduce the following idea. f of x, y be a function of two variables that's differentiable. The discriminant at some point is given by... Suppose our function has a critical value at the point. If our discriminant is positive, f has an extreme value. If our discriminant is negative, f has a saddle point. And if our discriminant is zero, the test is inconclusive. Note that at best, the discriminant will tell us that we have an extreme value and not whether it's a maximum or a minimum. So we already know the answer to this question, but let's go ahead and apply our discriminant. So we already found the critical value zero, zero. And we've already found the second partial derivative with respect to x and with respect to y. So we just need to find that mixed partial derivative, so we find... Now the second partial derivative with respect to x and with respect to y indicate that zero, zero is a minimum as long as it's an extreme value. So to decide whether we have an extreme value, we'll compute the discriminant. So everything depends on the sign of the discriminant. And since the discriminant is less than zero, the critical value corresponds to a saddle point and is neither a maximum nor a minimum. Let's try a different function. So we'll find our first partial derivatives. Again, we want to find the critical values. So we'll find when our derivatives are zero. So we have to solve our system of equations. Now we want to know whether we have a maximum or a minimum value. So we'll find our second derivatives as well as our mixed partial. We'll compute the discriminant. And since the discriminant is greater than zero, the critical value is an extreme value. But again, we don't know which one, whether it's a maximum or a minimum, but we do have this information about the second derivative. And since our second derivative with respect to x is greater than zero, the critical value does correspond to a minimum value.