Let f be continuous on [a,b] and differentiable on (a,b).
Then f(a) = f(b) ==> there exists a number, c, in (a,b) such that f'(c)=0.
If f is a constant function, then f'(c)=0 for all c in (a,b), proving this case.
If f(x) > f(a) for some x in (a,b) and c is a maximum of f on [a,b],
(c must exist, by the Extreme Value Theorem)
then f(c) ≥ f(x) > f(a) = f(b).
Since f(c) ≠ f(a), and f(c) ≠ f(b), it follows that c is not an endpoint of [a,b], so it is a relative maximum.
Since relative extrema occur only at critical numbers, c is a critical number of f, which means either f is not differentiable at c or f'(c)=0.
Well, f is differentiable at c, so f'(c)=0, proving this case.
Similarly if f(x) < f(a) for some x in (a,b) then let c be a minimum of f on [a,b]. This case is proved the same as above.
How is this theorem used?
A generalization of Rolle's Theorem is the Mean Value Theorem.
Extreme Value Theorem -- that a continuous function on a closed interval has a maximum (and a minimum).
Mean Value Theorem -- that f'(c) = (f(a)-f(b))/(a-b) for some c in (a,b)
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