Consider the function defined on by h(x)=|x|^ 1/3 (5-x)

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

Consider the function

h

defined on

\mathbb R

h(x)=|x|^{1/3}(5-x).

Throughout this problem, let

h'(x)

denote the derivative of

h

wherever it exists. (a) Determine the domain of

h

. State your reasoning clearly. (b) Identify all values of

x

where

h'(x)=0

and all values where

h'(x)

is undefined while

h

remains defined. (c) Describe, without computing a full sign chart, how the information from part (b) can be used to decide on which intervals

h

is increasing or decreasing. (d) Explain why

x=0

must be examined as a potential location of a relative extremum, even though

h'(0)

does not exist. Then decide whether

h

has a relative extremum at

x=0

and justify your conclusion.