Rolle's Theorem and a Proof

A special case of the Mean Value Theorem, known as Rolle's Theorem, is:

Rolle's Theorem If f(x) is continuous an [a,b] and differentiable on (a,b) and if f(a) = f(b) then there is some c in the interval (a,b) such that f '(c) = 0.

The reason that this is a special case is that under the stated hypothesis the MVT guarantees the existence of a point c with

		f(b) - f(a)		f(b) - f(a)
f '(c)	=		=		= 0.
		b - a		b - a

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Proof of Rolle's Theorem!

Most proofs in CalculusQuest^TM are done on enrichment pages. This is one exception, simply because the proof consists of putting together two facts we have used quite a few times already.
In the statement of Rolle's theorem, f(x) is a continuous function on the closed interval [a,b]. Hence by the Intermediate Value Theorem it achieves a maximum and a minimum on [a,b]. Either

1. One of these occurs at a point c with a < c < b,Since f(x) is differentiable on (a,b) and c is an extremum we then conclude that f '(c) = 0.
   or
   
2. Both the maximum and minimum occur at endpoints.Since f(a) = f(b), this means that the function is never larger or smaller than f(a). In other words, the function f(x) is constant on the interval [a,b] and its derivative is therefore 0 at every point in (a,b).

That's the proof! Truth in Proofs Statement

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