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# Derivatives

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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

*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.