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.
or
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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