Question

answer the following questions about the function whose derivative is $f(x)=(x - 2)^2(x + 6)$. a. what are the critical points of $f$? b. on what open intervals is $f$ increasing or decreasing? c. at what points, if any, does $f$ assume local maximum and minimum values? a. find the critical points, if any. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the critical point(s) of $f$ is/are $x=$ (simplify your answer. use a comm separate answers as needed.) b. the function $f$ has no critical points.

Explanation:

Step1: Find critical points

Critical points occur where \(f^{\prime}(x)=0\) or \(f^{\prime}(x)\) is undefined. Since \(f^{\prime}(x)=(x - 2)^{2}(x + 6)\) is a polynomial, it is defined for all real \(x\).
Set \(f^{\prime}(x)=0\):
\((x - 2)^{2}(x + 6)=0\)
Using the zero - product property \(ab = 0\Rightarrow a=0\) or \(b = 0\).
If \((x - 2)^{2}=0\), then \(x=2\). If \(x + 6=0\), then \(x=-6\).

Answer:

A. The critical point(s) of \(f\) is/are \(x=-6,2\)