A twice-differentiable function satisfies the following conditions: • for • for

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A twice-differentiable function

h

satisfies the following conditions: •

h''(x) > 0

for

x \in (-\infty, -2) \cup (1, 4)

•

h''(x) < 0

for

x \in (-2, 1) \cup (4, \infty)

(a) Describe the concavity of

h

on each of the four intervals determined by

x = -2

x = 1

, and

x = 4

. (b) Identify all

x

-values at which

h

may have a point of inflection and explain your reasoning. (c) Explain why a sign change in

h''

is necessary but not sufficient to guarantee an inflection point, citing any additional conditions that must be satisfied.