Question

answer the questions below about the function whose derivative is $f(x)=\frac{(x - 2)(x + 8)}{(x + 1)(x - 3)},x\
eq-1,3$.
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. what are the critical points of f? select the correct choice below and, if necessary, fill in the answer box within your choice.
○ a. $x=\square$ (use comma to 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.
Set \(f^{\prime}(x)=0\): \(\frac{(x - 2)(x + 8)}{(x + 1)(x - 3)}=0\). A fraction is zero when the numerator is zero (and denominator is non - zero). So, \((x - 2)(x + 8)=0\). Using the zero - product property \(x-2 = 0\) gives \(x = 2\) and \(x+8=0\) gives \(x=-8\).
The derivative \(f^{\prime}(x)\) is undefined when the denominator \((x + 1)(x - 3)=0\), i.e., \(x=-1\) or \(x = 3\). But critical points are in the domain of \(f(x)\). Assuming \(f(x)\) is differentiable (since we are given \(f^{\prime}(x)\) in the form of a rational function for \(x
eq - 1,3\)), the critical points are \(x=-8\) and \(x = 2\).

Answer:

A. \(x=-8,2\)