Mathematics

Problem

The function

g

is the first derivative of a differentiable function

f

. A verbal description of the graph of

g

is provided. •

g(x)=0

x=-3

x=0

, and

x=4

. •

g(x)<0

for

x<-3

. •

g(x)>0

for

-3<x<0

. •

g(x)<0

for

x<4

and

x>0

is a misprint. However, according to the original text, it should read:

g(x)<0

for

0<x<4

. •

g(x)>0

for

x>4

. •

g

is continuous everywhere. Based solely on this information about

g

, analyze the behavior of

f

. a. Part 1: (a) Determine the open intervals on which

f

is increasing and the open intervals on which

f

is decreasing. Explain your reasoning. b. Part 2: (b) Identify all

x

-coordinates of local extrema of

f

and state whether each is a local maximum or a local minimum. Justify each classification. c. Part 3: (c) Describe how the concavity of

f

might behave near

x=-3

x=0

, and

x=4

, even though no information about

g'

is given. Your answer should be qualitative rather than computational.

Intuitive explanation

Since $g$ is the derivative of $f$ , the zeros and sign of $g$ tell you exactly where $f$ has horizontal tangents and whether $f$ is increasing or decreasing. Where $g > 0$ , the function $f$ is climbing; where $g < 0$ , it's falling. The sign changes at $x = -3, 0, 4$ mark local extrema—use the sign pattern before and after each zero to determine whether each is a local maximum or minimum.

Intuitive explanation

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