﻿ Rolle's theorem

Rolle's theorem

The set of role ( named after the French mathematician Michel Rolle ) is a central theorem of differential calculus.

It indicates that a function f [b A ] at least one point of (a, b) the derivative is continuous and in the open interval (a, b ) is differentiable, and also satisfied in the closed interval zero.

This clearly means: On the graph of the function, there are between two curve points with matching y- values ​​of at least one point on the curve with a slope of m = 0, ie with a horizontal tangent. The theorem says so in particular, that is a zero of the derivative between two zeros of a differentiable function.

The set of role is both a special case of the mean value theorem of differential calculus. On the other hand, can be the mean value theorem with the help of Rolle's theorem to prove.

Evidence

For a constant function f the statement is trivial and applies to all c from the open interval (a, b).

It remains the case that f is not constant. There are then either x values ​​or with values ​​x with, or both. Since f [ b a] is ​​continuous with the compact interval, it takes ( on the Weierstrass' ) at a location c into [a, b], a maximum ( in the former case ) or a minimum ( in the latter case ) to. It must then be because a and b no maximum points ( in the first case ) or no minimum points ( in the second case ) can be.

At this extremal c must be the derivative equal to 0: Otherwise, we may without loss of generality assume that. Then for all x in a sufficiently small neighborhood of c if and when ( a direct consequence of the definition of f ' (c)), and there was then every neighborhood of x with c both elements as well as elements with x, in contradiction to the fact that f c has an extremum.

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