Consider the function

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

Problem

h(x)=\sin^{3}\!(2x)

. Provide detailed responses to the prompts below. (a) Compute

h'(x)

and simplify your result. (b) Explain why the critical points of

h

occur precisely when either

\sin(2x)=0

\cos(2x)=0

. (c) For

x

in the interval

[0,\pi]

, classify each critical point of

h

as a local maximum, local minimum, or neither. Support your classification with clear reasoning.