A special case of the Mean Value Theorem, known as Rolle's Theorem, is:
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|
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
Since f(x) is differentiable on (a,b) and c is an extremum we then conclude that f '(c) = 0.
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
All rights reserved---1996
William A. Bogley