Science:Math Exam Resources/Courses/MATH104/December 2015/Question 03 (c)/Solution 1

First, we plot our x-intercepts:

• $f(-5)=f(0)=0$

We then identify intervals of increase and decrease (see hint 1):

• $f'(x)<0$ for $-5<x<-3$ and $1<x<5$
• $f'(x)>0$ for $-3<x<1$

Next, we consider concavity (see hint 2):

• $f''(x)<0$ for $-2<x<3$
• $f''(x)>0$ for $-5<x<-2$ and $3<x<5$

Following these guidelines, we make a rough sketch of a possible function satisfying the above (see hint 3 for a tip):

Finally, we label local extrema and inflection points (where $f''(x)$ changes sign):