The extreme value theorem is also called the maximum and minimum value theorem, and its meaning is very intuitive: if the function f(x) is a continuous function in the interval [a, b], there must be a maximum value and a minimum value, and the maximum value and the minimum value are required at least once.

This is a very famous theorem, and the content of the theorem is intuitive and not difficult to understand. But it is not easy to prove it. It comes from interval set theorem and B-M theorem. This proof process is more complicated. Due to the limitation of space and level, this article can only skip this part, and interested students can learn about it themselves.

We assume that m and m are the minimum and maximum values of the function f(x) in the interval [a, b] respectively, then according to the extreme value theorem, we can get the following formula:

[formula]

This formula may seem a little complicated, but it is very simple after we draw the picture:

The shaded gray part in the above figure is the result of definite integral. The blue rectangular area is m(b-a) and the large rectangular area is M(b-a).

We can easily prove this conclusion by the relationship between geometric areas.

The mathematical proof is also very simple, because m and m are the minimum and maximum respectively, so the formula can be obtained. We regard the constant as a function, and by integrating it, we can get:

[formula]

The result of two-sided integration is a rectangular area, so we have proved it.

Integral mean value theorem

The extreme value theorem is simple, but it is the basis of many theorems, such as our integral mean value theorem, which is closely related to it.

Let's make a simple modification to the above formula. Because b-a is constant and greater than 0, we divide b-a on both sides of inequality [formula] at the same time, and we can get:

[formula]

We regard the formula as a whole, and its value is between the maximum and minimum of the function in the interval. According to the intermediate value theorem of continuous function, we can definitely find a point [formula] on [a, b] so that the value of f(x) at this point [formula] is equal to this value, namely:

[formula]

The above formula is the integral mean value theorem. There are two points to note here. Let's talk about a simple point first, that is, the mean value theorem of continuous functions we use. So it is limited that this must be a continuous function, otherwise it may just happen that the function is not defined at the [formula] point. This is also a prerequisite for the theorem to be established.

The second point is to briefly introduce the intermediate value theorem of continuous functions, which means that for a continuous function in the interval [a, b], for any constant between its maximum value and minimum value, a point must be found in the interval [a, b] so that the function value of this point is equal to this constant.