The extreme value of a function is the maximum or minimum value reached by the function in a certain area, and the point where the extreme value is obtained is called the extreme point. From the image of the function, the tangent of the extreme point is parallel to the X axis. The point where the tangent is parallel to the X axis is the extreme point of the function, and the same function can have multiple extreme points, which are not necessarily equal in size, so the extreme points may have the maximum and minimum values of the function, which are different from the maximum and minimum values of the function in a certain area.

The goal of mathematical optimization is to find the maximum and minimum in the whole function domain. If the function is continuous in the closed interval, there are maxima and minima in the whole region through the extreme value theorem. In addition, the maximum (or minimum) on the whole domain must be a local maximum (or minimum) within the domain, or it must be located on the boundary of the domain.

Therefore, the way to find the maximum (or minimum) in the whole domain is to look at all the local maximum (or minimum) inside, and also look at the maximum (or minimum) of points on the boundary, and take the maximum or minimum.