Chapter 2.3, Problem 74E

Solution

To determine the function represented in the graph from the given options.

The function given in the option $A)$ , $f\left(x\right)=x{\left(x+2\right)}^2\left(x-2\right)$ , is the most appropriate choice for the function represented in the graph.

Given the graph:

  

Explanation:

Observe the graph from the left side, it touches $x=-2$ and rises back to the positive side of $y$ -axis.

Then, it follows that, $x=-2$ is a zero of the function with multiplicity $2$ .

Now, the graph passes $x=0$ and $x=2$ directly without any complications.

That is, both $x=0$ and $x=2$ are zeroes of the function with the multiplicity $1$ .

Now, two choices left:

  $\begin{array}{l}A)f\left(x\right)=x{\left(x+2\right)}^2\left(x-2\right)\\ B)f\left(x\right)=x{\left(x+2\right)}^2\left(2-x\right)\end{array}$ .

From the graph, it is clear that $f\left(x\right)$ is negative at $x=1$ .

Now, consider the function $f\left(x\right)=x{\left(x+2\right)}^2\left(2-x\right)$ given in option $B)$ .

Substitute $x=1$ in $f\left(x\right)=x{\left(x+2\right)}^2\left(2-x\right)$ :

  $\begin{array}{l}f\left(1\right)=1\cdot {\left(1+2\right)}^2\left(2-1\right)\\ f\left(1\right)=9\end{array}$ .

Thus, $f\left(x\right)$ is positive for the function $f\left(x\right)=x{\left(x+2\right)}^2\left(2-x\right)$ given in option $B)$ .

That is, option $B)$ is not an appropriate choice of the function represented in the graph.

Then, it follows that, the function given in the option $A)$ , $f\left(x\right)=x{\left(x+2\right)}^2\left(x-2\right)$ , is the most appropriate choice for the function represented in the graph.

Conclusion:

The function given in the option $A)$ , $f\left(x\right)=x{\left(x+2\right)}^2\left(x-2\right)$ , is the most appropriate choice for the function represented in the graph.