A twice-differentiable function has a second derivative whose sign changes as…

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

Problem

A twice-differentiable function

f

has a second derivative

f''

whose sign changes as follows: •

f''(x)>0

for

x<-3

. •

f''(-3)=0

. •

f''(x)<0

for

-3<x<1

. •

f''(1)=0

. •

f''(x)<0

for

1<x<4

. •

f''(4)=0

. •

f''(x)>0

for

x>4

. No other zeros of

f''

exist. Assume

f

and

f'

are continuous everywhere. (a) Determine the intervals on which the graph of

f

is concave up and the intervals on which it is concave down. Explain your reasoning. (b) Identify all

x

-values at which the graph of

f

has a point of inflection. Justify each value you state.