The Mean Value Theorem
The Mean Value Theorem is a glorified version of Rolle's Theorem.
The Mean Value Theorem states:
If f is continuous on [a, b], and f is differentiable on (a, b), then there is some c in (a, b) with
f ' (c) = [f(b) - f(a)] / [b - a].
[f(b) - f(a)] / [b - a] is the slope of the secant line between the points (a, f(a)) and (b, f(b)).
As with the Intermediate Value Theorem and Rolle's Theorem, the conclusion of the Mean Value Theorem leaves a lot of things out. It doesn't tell us the value of c where f'(c) equals the slope of the secant line. It doesn't tell us how many such values of c there are. All the Mean Value Theorem does is guarantee that if all the hypotheses are met, at least one such c exists.
As with all the other theorems, if the necessary hypotheses aren't met we then aren't allowed to use the Mean Value Theorem.
However, for any c in (-1,1) f ' (c) is either positive (if c < 0) or negative (if c > 0). There is no point c in (-1, 1) with f ' (c) = 0.
However, the derivative of f is either -1, +1, or undefined:
There is no value of c in (-1, 1) (or anywhere) with f'(c) = 0.