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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 

' (c) = [f(b) - f(a)] / [b - a]. 

[f(b) - f(a)] / [b - a] is the slope of the secant line between the points (af(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) ' (c) is either positive (if c < 0) or negative (if c > 0). There is no point c in (-1, 1) with ' (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.

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