Science:Math Exam Resources/Courses/MATH110/April 2017/Question 06 (c)/Solution 1

We will make a table outlining the information from part (a):

${\begin{array}{c|c|c|c|c|c|c|c|c}x&x<-5&-5<x<-3&-3<x<-2&-2<x<0&0<x<2&2<x<3&3<x<5&x>5\\f'(x)&+&+&-&+&+&-&+&+\\f''(x)&+&-&-&-&+&+&+&-\end{array}}$

We will use this to create a similar table for f:

${\begin{array}{c|c|c|c|c|c|c|c|c}x&x<-5&-5<x<-3&-3<x<-2&-2<x<0&0<x<2&2<x<3&3<x<5&x>5\\f(x)&{\text{inc}}&{\text{inc}}&{\text{dec}}&{\text{inc}}&{\text{inc}}&{\text{dec}}&{\text{inc}}&{\text{inc}}\\f(x)&{\text{CCU}}&{\text{CCD}}&{\text{CCD}}&{\text{CCD}}&{\text{CCU}}&{\text{CCU}}&{\text{CCU}}&{\text{CCD}}\end{array}}$

Note that this is consistent with inflection points at -5, 0, and 5, and with the fact that -3 is a maximum and 3 is a minimum.

We put the pieces together and arrive at the graph below: