A twice-differentiable function satisfies the following conditions

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

Problem

A twice-differentiable function

f

satisfies the following conditions. •

f

is decreasing for

x < 1

and increasing for

x > 1

. •

f

has a relative minimum at

x = 1

. • The slope of the tangent line to the graph of

f

0

x = 1

and

4

x = 5

. •

f

is concave upward on

(-\infty,0)

and

(2,\infty)

and concave downward on

(0,2)

. (1) Part 1: On the coordinate axes provided on your answer sheet, sketch one possible graph of

f

that satisfies all of the stated conditions. Clearly indicate the behavior near

x = 0

x = 1

, and

x = 2

. (2) Part 2: Based only on your sketch or on direct reasoning from the conditions above, sketch a possible graph of

f'

on the same set of axes and label it. (3) Part 3: Similarly, sketch a possible graph of

f''

and label it. (4) Part 4: Describe verbally how the placement, sign, and shape of your three sketches illustrate the relationships among

f

f'

, and

f''

for

0 < x < 2