Consider a twice–differentiable function defined on whose graph satisfies the…

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

Problem

Consider a twice–differentiable function

p

defined on

-6\le x\le 6

whose graph satisfies the following conditions. •

p

is decreasing and concave down on

[-6,-2]

. •

p

is decreasing and concave up on

(-2,0)

. •

p

attains an absolute minimum at

x=0

. •

p

is increasing and concave up on

(0,3)

. •

p

is increasing and concave down on

(3,6]

. • The slopes of the tangent lines at

x=-4,\,x=-1,\,x=1,\,

and

x=5

are

-3,-1,2,

and

1

, respectively. Throughout this problem,

p'

denotes the first derivative of

p

. (a) Construct a complete sign chart for

p'(x)

on the interval

-6\le x\le 6

, indicating all intervals on which

p'(x)

is positive, negative, or zero. (b) Using your sign chart from part (a) and the numerical slope information provided, sketch a possible graph of

y=p'(x)

for

-6\le x\le 6

. Your sketch should be qualitatively accurate with respect to intercepts, relative extrema, and concavity. (c) Explain how the concavity changes of

p

specified in the prompt are reflected in the behavior of

p'

in your graph from part (b).