Basic Analysis I — Real Analysis · Problem 4.1.16

The Derivative — Jiri Lebl

Mathematics — Real Analysis · derivative-sign-change

Problem

Suppose

f : (a,b) \to \mathbb{R}

is differentiable at

c \in (a,b)

f(c) = 0

, and

f'(c) > 0

. Prove that there is a

\delta > 0

such that

f(x) < 0

whenever

c - \delta < x < c

and

f(x) > 0

whenever

c < x < c + \delta