A twice-differentiable function is defined for

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

A twice-differentiable function

f

is defined for

-2 \le x \le 6

. Let

g

denote its derivative, so

g(x)=f'(x)

. A verbal description of the graph of

g

is provided below. •

g

is continuous on the entire interval. •

g

crosses the

x

-axis at

x=0

and

x=4

. •

g

attains a local maximum value of

3

x=1

and a local minimum value of

-2

x=3

. • For

x<0

g

is decreasing; on

(0,1)

it is increasing; on

(1,3)

it is decreasing; on

(3,4)

it is increasing; and for

x>4

it is decreasing. Assume there are no other critical points of

g

besides those listed. (a) Identify the

x

-coordinates of all critical points of

g

on the interval

-2 \le x \le 6

and classify each as a local maximum, local minimum, or neither. Justify your classifications. (b) Determine the open intervals on which the graph of

f

is concave upward and the open intervals on which it is concave downward. Give a clear justification based on the behavior of

g

. (c) Indicate all

x

-coordinates at which the graph of

f

has a point of inflection. Explain why each point you list satisfies the definition of an inflection point. (d) Suppose

f(0)=5

and

f(2)=3

. Describe how the concavity you found in part (b) constrains the possible shape of the graph of

f

0\le x\le4

. Include a discussion of the relative location of the point

(2,3)

with respect to the line segment joining

(0,5)

and

(4,f(4))

. No explicit sketch is required, but your reasoning must be articulated clearly.