1549901369-Elements_of_Real_Analysis__Denlinger_

318 Chapter 6 • Differentiable Functions

Proof. Exercise 1. •

Theorem 6.3.4 (Local Extreme Value Theorem) If a function f has a

local extreme value at an interior point xo of its domain, then either f'(xo) = 0

or f' ( xo) does not exist.

Proof. We shall prove the contrapositive. Suppose it is not true that either

f'(xo) = 0 or f'(xo) does not exist. That is, f'(xo) exists and f'(xo)-:/:-0. Then

either f'(xo) > 0 or f'(xo) < 0.

Case 1 (f'(x 0 ) > 0): By Theorem 6.3.2, :J r5 > 0 3

(a) '<Ix E (xo - r5, xo), f(x) < f (xo), and

(b) '<Ix E (xo, xo + r5), f(x) > f (xo).

By (a), f cannot have a local minimum at xo. By (b), f cannot have a local

maximum at x 0. Since f has neither a lo cal maximum nor a local minimum at

x 0 , it does not have a local extreme value at x 0.

Case 2 (f'(xo) < 0): Exercise 2. •

MONOTONE FUNCTIONS

Monotone functions were defined in Section 5.2 (see Definition 5.2.15). As

you recall from elementary calculus, there is a natural relationship between the

monotonicity of a function and the sign of its derivative. We shall now explore

that relationship.

Theorem 6.3.5 (a) If f is differentiable at x 0 and monotone (or strictly) in-

creasing on an open interval I containing x 0 , then, f' ( x 0 ) 2". 0.

(b) If f is differentiable and monotone (or strictly) decreasing on an open

interval I containing xo, then, f' ( xo) :S 0.

Proof. (a) Suppose f is differentiable at x 0 and monotone increasing on

an open interval I containing x 0. Then '<Ix< x 0 in I,

f (x) :S f (xo), so

f(x) - f(xo) :S 0 and x - Xo < 0, so

f(x) - f(xo)

-----2".0.

x -xo

Thus, since limits from the left preserve inequalities, lim f(x) - f(xo) > O.
x-+x 0 X - Xo -
Since f is differentiable at xo, this limit (from the left) exists and equals f'(x 0 ).
Therefore, f'(xo) 2". 0.

(b) Exercise 3. •