I. Minimum value

In mathematical analysis, in a given range (relative extreme value) or the whole definition domain of a function (global or absolute extreme value), the maximum and minimum values of a function are collectively called extreme values (poles). PierredeFermat was one of the first mathematicians to put forward the maximum and minimum values of functions.

According to the definition of set theory, the maximum value and minimum value of a set are the largest element and the smallest element in the set respectively. An infinite set, a set of real numbers, with no minimum and maximum.

II. Introduction

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

So the way to find the global maximum (or minimum) is to look at all the local maximum (or minimum) inside, and also look at the maximum (or minimum) of the points on the boundary, and take the maximum or minimum. Fermat's theorem can find the differential function of local extreme value, which shows that it must happen at the critical 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.

Third, multivariable functions.

Similar conditions apply to multivariable functions. For example, in the following (enlarged) figure 2, the necessary condition of local maximum is similar to that of a function with only one variable. The maximum value of the first partial derivative about z (the variable to be maximized) is zero (the luminous point at the top of Figure 2).

The second partial derivative is negative. Because saddle points may exist, these are only necessary conditions for local maxima. In order to use these conditions to solve the maximum value, the function z must also be distinguishable. The second-order partial derivative test can help to classify points as relative maxima or relative minima.