The derivative of a twice-differentiable function has the following properties: •…

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

Problem

The derivative

g = f'

of a twice-differentiable function

f

has the following properties: •

g(x) = 0

exactly at

x = -1

x = 2

, and

x = 5

. •

g(x) < 0

for

x < -1

and for

2 < x < 5

. •

g(x) > 0

for

-1 < x < 2

and for

x > 5

. •

g

is decreasing on

(-1, 2)

and increasing on

(2, 5)

. Suppose

f(0) = 3

. (a) Determine the

x

-coordinates of all relative extrema of

f

and classify each as a relative maximum or relative minimum. Provide a complete justification. (b) Determine the sign of

f''(0)

. Explain your reasoning in terms of the slope of

g

. (c) Without computing exact values, rank the four numbers

f(-1)

f(0)

f(2)

, and

f(5)

from least to greatest. Support your ranking with clear justification based on the behavior of

g