1549901369-Elements_of_Real_Analysis__Denlinger_

6.3 Local Extrema and Monotone Functions 317

Theorem 6.3.2 Suppose f'(xo) > 0 at an interior point xo of V(f). Then
:Jo>03

(a) Vx E (xo - o, xo), f(x) < f (xo), and

(b) Vx E (xo, xo + o), f(x) > f (xo). (See Figure 6.4-)

Proof. Suppose f'(xo) > 0 at an interior point xo E V(f). Then

lim f(x) - f(xo) > 0.

x->xo x - Xo

By the "bounded away from zero" theorem^7 for limits of functions, :lo >

0 3 0 < Ix -xol < o::::? f(x) - f(xo) > 0. Then,
x -xo
(a) XE (xo -O,xo) ::::? X - Xo < 0 and f(x) - f(xo) > 0
x - xo
::::? f(x) - f(xo) < 0
::::? f(x) < f(xo).
f(x) - f(xo)
(b) X E (xo, Xo + o) ::::? X - Xo > 0 and > 0
x - xo
::::? f(x) - f(xo) > 0
::::? f(x) > f(xo). •

y

x

Figure 6.4 Figure 6.5

x

Theorem 6.3.3 Suppose f'(x 0 ) < 0 at an interior point xo of V(f). Then
:io>03

(a) Vx E (xo - o, xo), f(x) > f (xo), and

(b) Vx E (xo, xo + o), f(x) < f (xo).

7. See Definition 4.2.8 and Theorem 4.2.9.

(See Figure 6.5.)