Mathematics

Problem

Let

g

be a twice–differentiable function with the following properties on the interval

0\le x\le 6

. •

g''(x)>0

for

0<x<4

. •

g''(x)<0

for

4<x<6

. • The graph of

y=g''(x)

and the

x

–axis enclose a total signed area of

12

square units on

0\le x\le 4

and

-9

square units on

4\le x\le 6

(area below the axis is taken to be negative). •

g'(0)=5

. Answer the following questions. Show all work leading to your answers. (a) Compute

g'(4)

and

g'(6)

. Indicate units of measure, if any. (b) Based on your results in part (a), determine on which subintervals of

[0,6]

the function

g'

is increasing and on which it is decreasing. (c) Using the sign of

g''

together with the behavior of

g'

, identify the interval(s) on which the graph of

g

is concave up and the interval(s) on which it is concave down. (d) Decide whether

x=4

x=6

is a point of inflection for the graph of

g

. Fully justify your conclusion.