Chapter 2.2, Problem 10Q

To determine

The $x-\text{intercept}$ of the graph of the function and decide where the function is increasing and decreasing.

Answer to Problem 10Q

  $x=-2$ & $x=-4$ are the required $x-\text{intercept}$ of the function $g\left(x\right)=-3\left(x+2\right)\left(x+4\right)$ and the function is increasing in $\left(-\infty ,-3\right]$ and decreasing in $\left[-3,\infty \right)$ .

Explanation of Solution

Given information : Given function is $g\left(x\right)=-3\left(x+2\right)\left(x+4\right)$ .

To find the $x-\text{intercept}$ of the function $g\left(x\right)=-3\left(x+2\right)\left(x+4\right)$ , we need to solve the equation $g\left(x\right)=0$ .

Therefore,

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

Thus, $x=-2$ & $x=-4$ are the required $x-\text{intercept}$ of the function $g\left(x\right)=-3\left(x+2\right)\left(x+4\right)$ .

To decide where the function is increasing and decreasing, we need to plot the graph of $g\left(x\right)=-3\left(x+2\right)\left(x+4\right)$ .

  

Clearly, from the above graph, the function is increasing in $\left(-\infty ,-3\right]$ and decreasing in $\left[-3,\infty \right)$ .