The definition function is defined in a neighborhood of a point, if all points different from the neighborhood are suitable for inequality.

It is said that this function has a maximum value at point. If they are all suitable for inequality

Then say that this function has a minimum at some point. The maximum and minimum values are collectively called extreme values. The point at which the function obtains the extreme value is called the extreme value point.

Example 1 function has a minimum at point (0,0). Because for any point different from (0,0) in the neighborhood of point (0,0), the function value is positive, and the function value at point (0,0) is zero. Geometrically, this is obvious because the point (0,0,0) is the vertex of an elliptic paraboloid with an upward opening.

Example 2 This function has a maximum value at point (0,0). Because the function value at point (0,0) is zero, but for any point in the neighborhood of point (0,0) that is different from (0,0), the function value is negative, and point (0,0,0) is the vertex of the cone under the plane.

The function of Example 3 has neither a maximum nor a minimum at point (0,0). Because the function value at point (0,0) is zero, there are always points that make the function value positive and points that make the function value negative in any neighborhood of point (0,0).

Theorem 1 (necessary condition) If a function has a partial derivative at a point and an extreme value at a point, then its partial derivative at that point must be zero:

The certificate can be set at the point with the maximum value. According to the definition of maximum, all different points in a certain neighborhood of a point are suitable for inequality.

In particular, the point of "sum" in this neighborhood should also be suitable for inequality.

This means that the unary function gets the maximum value at, so it must be.

It can also prove that

Geometrically, if a surface has a tangent plane at this point, then this tangent plane.

Become a plane parallel to the coordinate plane.

According to the theorem 1, the extreme point of a function with partial derivatives must be the stagnation point. But the stagnation point of the function is not necessarily the extreme point. For example, the point (0,0) is the stagnation point of the function, but the function has no value at this point.

How to determine whether the stagnation point is an extreme point? The following theorem answers this question.

Theorem 2 (Sufficient Condition) Let a function be continuous in a certain neighborhood of a point and have first and second continuous partial derivatives, and let

The conditions for obtaining the extreme value are as follows:

(1) has an extreme value, a maximum value at that time and a minimum value at that time;

(2) There is no extreme value;

(3) There may or may not be an extreme value, which needs to be discussed separately.

This theorem has not been proved now. Using the theorem 1 and 2, we describe the solution of the extreme value of a function with second-order continuous partial derivatives as follows: