1549901369-Elements_of_Real_Analysis__Denlinger_

316 Chapter 6 • Differentiable Functions

(c) if f is continuous on [a, b) and lim f(x) = f(a), then Jis continuous

x---+b-

on R

(d) if f is differentiable on (a, b), differentiable from the right at a, and

lim f' ( x) = lim f' ( x), then j is differentiable on R

X--tb- X--ta+

6.3 Local Extrema and Monotone Functions

In this section, we state and prove theorems that justify the procedures used

in elementary calculus courses to find local maximum and minimum values of

functions using their derivatives.

Definition 6.3.1 Suppose f is defined in a neighborhood of x 0. Then
(a) f has a local maximum at xo if, for some neighborhood of xo, f takes on
its maximum value at x 0. That is,
:JO > 0 3 f (xo) =max f (N 0 (xo)); equivalently,
:lo> 0 3 '\:/x E N 0 (xo) n V(f), f(x) :::; f(xo).
(b) f has a local minimum at x 0 if, for some neighborhood of x 0 , f takes on
its minimum value at x 0. That is,
:Jo> 0 3 f(xo) = minf (No(xo)); equivalently,
:Jo> 0 3 '\:/x E N 0 (xo) n V(f), f(x) ~ f(xo).
(c) a function f has a local extreme value at x 0 if it has either a local
maximum or a local minimum at x 0.

y

,r-Maxf(N 0 (x 0 ))

I I I I

Figure 6.3

x

The next theorem is the basic tool needed to prove that if a differentiable
function f has a local extreme point at an interior point x 0 E V(f), then
f'(xo) = 0.