A function is twice differentiable for

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

Problem

A function

f

is twice differentiable for

-5<x<5

. The sign of

f''(x)

on this interval is summarized in the table below.

Interval	$(-5,-1)$	$(-1,0)$	$(0,3)$	$(3,5)$
Sign of $f''(x)$	$+$	$-$	$-$	$+$

Answer the following questions about the concavity of

f

. (a) Identify the open intervals on which the graph of

f

is concave up and the open intervals on which it is concave down. (b) The points

x=-1

x=0

, and

x=3

are the only values in

(-5,5)

where

f''(x)=0

f''(x)

does not preserve a single sign. For each of these three

x

–values, state whether

x

corresponds to a point of inflection of the graph of

f

and provide a clear justification based on the information in the table. (c) Suppose further that

f(-1)=4

and

f(3)=10

. Give the exact coordinates of every point of inflection of the graph of

f

in the interval

-5<x<5

. Explain your reasoning.