Chapter 2.8, Problem 65E

Solution

To find: The graph of the given function changes sign at $x=0$ .

The answer is false.

Given information:

Consider the given function.

  $f\left(x\right)=x^4{\left(x+3\right)}^2{\left(x-1\right)}^3$

Calculation:

First find the boundary points of the function $f\left(x\right)$ .

Substitute $f\left(x\right)=0$ and calculate the values of $x$ :

  $\begin{array}{c}{x}^4{\left(x+3\right)}^2{\left(x-1\right)}^3=0\\ x^4=0\\ x=0\end{array}$

And

  $\begin{array}{c}{x}^4{\left(x+3\right)}^2{\left(x-1\right)}^3=0\\ {\left(x+3\right)}^2=0\\ x+3=0\\ x=-3\end{array}$

And

  $\begin{array}{c}{x}^4{\left(x+3\right)}^2{\left(x-1\right)}^3=0\\ {\left(x-1\right)}^3=0\\ x-1=0\\ x=1\end{array}$

So, the boundary points of the function $f\left(x\right)$ are $-3,0,1$ .

Now, locate these zeros on the number line (or sign chart) and dividing the number line into intervals and check the function $f\left(x\right)$ is positive or negative at each interval below:

  

Therefore, the sign changes of the function $f\left(x\right)$ at $x=1$ .

Hence, the answer is false.