6.3 Local Extrema and Monotone Functions 319Note: In the next section, after we have proved the mean value theorem,
we shall prove a partial converse of this theorem.Caution: Taken together, Theorems 6.3.2- 6.3.5 might lead you to con-
clude erroneously that if f'(x 0 ) > 0, then f must be monotone increasing on
some neighborhood of x 0 , and if f'(x 0 ) < 0, then f must be monotone decreas-
ing on some neighborhood of x 0. Indeed, your intuition may strongly suggest
this. The following example will show that such a conclusion is not necessarily
true.
Example 6.3.6 The function^8 f(x) = { x +^2 x
2
sin(~) ~f x =I= O} is differen-
0 if x = 0
tiable everywhere, f'(O) > 0, but f is not monotone on any neighborhood of
0.
Proof. See Exercise 7. 0*INTERMEDIATE VALUE PROPERTY OF DERIVATIVES
Suppose f is differentiable on I= [a, b], where a< b. Then f is continuous
on [a, b]. The derived function f'(x) exists on [a, b] but is not necessarily con-
tinuous there. However, f' does have a very remarkable property in common
with continuous functions: the intermediate value property.
Theorem 6.3.7 (Intermediate Value Property of Derivatives) Suppose
f is differentiable on an open interval containing a and b, where a < b. If k is
any number between f'(a) and f'(b), then :Jc E (a,b) 3 f'(c) = k.
*Proof. Suppose f is differentiable on an open interval I containing a and
b, where a < b, and k is any number b etween f'(a) and f'(b). Then either
f'(a) < k < f'(b) or f'(a) > k > f'(b).
Case 1 (f'(a) < k < f'(b)): Define the function g on I by
g(x) = kx - f(x).
Then g is continuous on [a, b], so by the extreme value theorem (5.3.7), it
has a maximum value on [a, b]. Observe that
(a) g'(a) = k - f'(a) > 0, and by hypothesis, a is an interior point of D(g).
Thus, by Theorem 6.3.2, g cannot have its maximum value for [a, b] at a.
(b) g'(b) = k - f'(b) < 0, and by hypothesis, bis an interior point of D(g).
Thus, by Theorem 6.3.3, g cannot have its maximum value for [a, b] at b.
- This example comes from [49] Gelbaum and Olmsted, Counterexamples in Analysis- a
wonderful source of examples.