The Art and Craft of Problem Solving

332 CHAPTER 9 CALCULUS

a u b

Rolle's theorem has an important generalization, the mean value theorem.

If f(x) is continuous on [a , b] and differentiable on (a,b) , then there is

apointuE (a,b) at which

f'(u) =

f(b�=�(a)

.

In geometric terms, the mean value theorem asserts that there is an x-value u E (a,b)

at which the slope of the tangent line at (u,j(u)) is parallel to the secant line joining

(a,j(a)) and (b,j(b)). And the proof is just one sentence:

Ti lt the picture fo r Rolle 's theorem!

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The mean value theorem connects a "global" property of a function (its average rate of

change over the interval [a , b] ) with a "local" property (the value of its derivative at a

specific point) and is thus a deeper and more useful fact than is apparent at first glance.

Here is an example.

Example 9.3.4 Suppose f is differentiable on ( -00, 00 ) and there is a constant k < 1

such that If' (x) I :S k for all real x. Show that f has a fixed point.

Solution: Recall from Example 9.2 .4 on page 324 that a fixed point is a point x

such that f(x) = x. Thus we must show that the graphs of y = f(x) and y = x will

intersect. Without loss of generality, suppose that f(O) = v > 0 as shown.