A twice-differentiable function is defined for all real numbers

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

Problem

A twice-differentiable function

h

is defined for all real numbers. It is known that

h''(x)>0

for

x\in(-\infty,-1)\cup(2,6)

and

h''(x)<0

for

x\in(-1,2)\cup(6,\infty)

. In addition,

h''(-1)=h''(2)=h''(6)=0

. (a) Determine the intervals on which the graph of

h

is concave up and the intervals on which it is concave down. Express each answer using interval notation. (b) Explain how the given information about

h''

determines where

h'

is increasing and where

h'

is decreasing. (c) Suppose

h'(0)=3

and

h'(4)=-2

. Explain whether these values for

h'

are consistent with the concavity information provided, and justify your conclusion.