The graph of a continuous function possesses the following qualitative features

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

Problem

The graph of a continuous function

g

possesses the following qualitative features. •

g

is strictly decreasing on

(-\infty,-1)

and has a sharp corner at

x=-1

. • On

(-1,3)

the graph is smooth and strictly increasing, with a point of inflection at

x=1

. • At

x=3

the graph attains a smooth local maximum, then decreases smoothly on

(3,\infty)

. Use this information to investigate the behavior of the first derivative

g'

. (a) Describe the sign of

g'(x)

on each of the following intervals:

(-\infty,-1)

(-1,1)

(1,3)

, and

(3,\infty)

. Provide brief justification for each interval. (b) Identify every

x

–value at which

g'(x)=0

g'(x)

does not exist. Classify each such point of

g

as a local minimum, local maximum, or neither, and justify your classification. (c) Explain why

g'

is not defined at

x=-1

, even though

g

is continuous there. (d) Produce a qualitative sketch of

y=g'(x)

that is consistent with your analyses in parts (a)–(c). Indicate all zeros of

g'

and any points of discontinuity.