1, minimum value
Let the domain of the function y=f(x) be I, and if there is a real number m, it satisfies the requirement that for any real number x∈I, there is f(x)≥M and x0 ∈ i. Let f(x0)=M, then we call the real number m the minimum value of the function y=f(x).
2. Maximum value
Let the domain of the function y=f(x) be I, and if there is a real number m, it satisfies that for any real number x∈I, f(x)≤M and x0 ∈ i. Let f(x0)=M, then we call the real number m the maximum value of the function y=f(x).
Maximum value problem of function
On the function f: a->; R, if aEA exists, repair)
The maximum and minimum values are collectively referred to as absolute extremum or global extremum. If the maximum (minimum) value of a function exists, it must be unique, but the corresponding maximum (minimum) value point is not necessarily unique. A continuous function on the bounded closed set of r "must have a maximum and a minimum. This is the main basis for judging whether a function has an absolute extreme value.
In order to find the maximum and minimum, the basic method is to determine their existence first, and then compare the function values of functions at stagnation points, boundary points or nondifferentiable points, in which the maximum (small) value is the maximum (small) value. In many application problems, the existence of maximum and minimum can often be determined by the background of specific problems.
Fermat was the first person to find the maximum and minimum by differential calculus. He discovered that the necessary condition of extreme value is called Fermat theorem (not the present form), and determined that the function reaches the maximum or minimum value at the stagnation point. The extreme value problem has always been a concern of mathematicians, and several mathematical disciplines study complex extreme value problems, such as convex analysis, mathematical programming, variational methods and so on.