The derivative of a differentiable function is represented by the following verbal…

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

Problem

The derivative

f'(x)

of a differentiable function

f

is represented by the following verbal description of its graph: • The graph of

f'

is a continuous, piecewise-linear curve on

-6\le x\le 8

. • It crosses the

x

-axis at

x=-1

and

x=3

only. • On

(-6,-1)

the graph of

f'

lies entirely above the

x

-axis. • Between

x=-1

and

x=3

the graph of

f'

lies entirely below the

x

-axis. • On

(3,8)

the graph of

f'

is again entirely above the

x

-axis. No other features (such as vertical asymptotes or cusps) occur in this interval. Answer the following questions about

f

on the interval

-6<x<8

. (a) Determine the open intervals on which

f

is increasing and the open intervals on which

f

is decreasing. (b) Explain how the information in the description of

f'

justifies your answer in part (a). (c) Identify the

x

-coordinates of all relative extrema of

f

and classify each as a relative minimum or a relative maximum. Justify your classifications.