What is "mean value theorem"

n. (context calculus English) a statement that claims that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the average derivative of the arc.

In mathematics, the mean value theorem states, roughly, that given a planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.

The theorem is used to prove global statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.

More precisely, if a function f is continuous on the closed interval [a, b] , where a < b , and differentiable on the open interval (a, b) , then there exists a point c in (a, b) such that:

$$f'(c)=\frac{f(b)-f(a)}{b-a}.$$

It is one of the most important results in differential calculus, as well as one of the most important theorems in mathematical analysis, and is useful in proving the fundamental theorem of calculus. The mean value theorem follows from a more specific statement of Rolle's theorem, and can be used to prove the most general statement of Taylor's theorem (with Lagrange form of the remainder term).

In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives.