Chapter 5.3, Problem 3QQ

To determine

To find : which of the option is true.

Answer to Problem 3QQ

Option B is correct.

Explanation of Solution

Given information :

If g is a differentiable function such $g\left(x\right)<0$ that for all numbers x, and if $f\text{'}\left(x\right)=\left(x^2-9\right)g\left(x\right)$ .

Calculation:

The given function is $f\text{'}\left(x\right)=\left(x^2-9\right)g\left(x\right)$ .

Find critical point by setting $f\text{'}\left(x\right)=0$ and solve for x:

  $\begin{array}{l}\left(x^2-9\right)g\left(x\right)=0\\ \left(x^2-9\right)=0\\ \left(x-3\right)\left(x+3\right)=0\\ x-3=0\text{ or }x+3=0\\ x=3\text{ or }x=-3\end{array}$

The critical point partition the x-axis into three intervals as shown below:

  

From the first derivative test the sign of $f\text{'}$ changes its from negative to positive at $x=-3$ and $f\text{'}$ changes its from positive to negative at $x=3$

Therefore, there is relative maximum at $x=3$ and relative minimum at $x=-3$ .

Hence,

Option B is correct.