The necessary and sufficient conditions for the extreme value of a function are as follows: ① There is a point where the derivative is equal to 0, that is, at this point f' = 0. (2) Make the derivative equal to the x value of 0, and the signs of the left and right derivatives are opposite. if f ' left >； 0, f' right 0, is the minimum value.

In mathematical analysis, the maximum and minimum values (maxima and minima) of a function are collectively called extremum (poles), and extremum is the maximum and minimum values (local or relative extremum) of a function within a given range or the global (global or absolute extremum) of a function. Pierre de Fermat was one of the first mathematicians to discover 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. Infinite infinite set, real number set, no minimum and maximum.

Extreme value is the maximum or minimum value of a function. If a function has a value everywhere in the neighborhood of a point, and the value of that point is the maximum (minimum), then the value of this function at that point is the maximum (minimum). If it is greater than (less than) the function values of other points in the neighborhood, it is a strict maximum (less than). This point is correspondingly called extreme point or strict extreme point.