Basic Analysis I — Real Analysis · Problem 4.3.11

The Derivative — Jiri Lebl

Mathematics — Real Analysis · second-derivative-test-general

Problem

Prove the more general version of the second derivative test. Suppose

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

is differentiable and

x_0 \in (a,b)

is such that,

f'(x_0) = 0

f''(x_0)

exists, and

f''(x_0) > 0

. Prove that

f

has a strict relative minimum at

x_0

. Hint: Consider the limit definition of

f''(x_0)