A differentiable function has a derivative

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

Problem

A differentiable function

f

has a derivative

g

, where

g = f'

. A verbal description of the graph of

g

is given below. •

g(x) < 0

for

x < -3

. •

g(x) = 0

x = -3

. •

g(x) > 0

for

-3 < x < 1

. •

g(x) = 0

x = 1

. •

g(x) < 0

for

1 < x < 5

. •

g

decreases on

(-\infty, -1)

, increases on

(-1, 3)

, and then decreases again on

(3, 5)

. Answer the following, showing clear reasoning in each case. (1) Part 1: (a) Determine the intervals on which

f

is increasing and the intervals on which

f

is decreasing. (2) Part 2: (b) Identify each

x

-coordinate at which

f

attains a relative maximum or a relative minimum on the open interval

(-\infty, 5)

. Classify each critical point and justify your classification. (3) Part 3: (c) Determine the intervals on which the graph of

f

is concave up and the intervals on which it is concave down. (4) Part 4: (d) State all

x

-coordinates at which the graph of

f

has a point of inflection and give a brief reason for each.