Chapter 3.5, Problem 26E

To determine

To graph: $f\left(x\right)$ and find the interval(s) at which the function is increasing and decreasing respectively.

Answer to Problem 26E

  

The function is decreasing on $\left(-3,1\right)$ and increasing on $\left(-\infty ,-3\right)$ and $\left(1,\infty \right)$ .

Explanation of Solution

Given information:

  $y=x^3+3x^2-9x$

Calculation:

Solve the function for x by factorizing $x^3+3x^2-9x$ as follows:

  $\begin{array}{l}{x}^3+3x^2-9x=0\\ x\left(x^2+3x-9\right)=0\\ x{\left(x-3\right)}^2=0\\ x=0,\frac{\pm 3\sqrt{5}-2}{2}\end{array}$

From the end behavior, the graph must start low and end high, because the highest power is odd and positive.

  

From the graph, it is clear that it is decreasing on $\left(-3,1\right)$ and increasing on $\left(1,\infty \right)$ and $\left(-\infty ,-3\right)$ .