Extreme values are defined as follows:
The concept of extreme value comes from the problem of maximum and minimum value in mathematical application. According to the law of extremum, every continuous function defined in a bounded closed region must reach its maximum and minimum. The problem is to determine at which point it reaches the maximum or minimum.
If the extreme point is not a boundary point, then it must be an interior point. Therefore, the first task here is to find a necessary condition for an interior point to become an extreme point.
Extended data:
Solve the extreme value of a function
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.
Fermat's last theorem can find the differential function of local extreme value, which means it must happen at a key point. We can use the first derivative test, the second derivative test or the higher derivative test to distinguish whether the critical point is a local maximum or a local minimum, and give sufficient distinguishability.
For any function defined by a segment, find the maximum (or minimum) of each part separately, and then find out which is the maximum (or minimum) to find the maximum (or minimum).
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