Question

answer the following questions about the function whose derivative is $f(x)=\frac{x^{2}(x - 2)}{x + 5}$, $x\
eq - 5$.
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 or minimum values?
a. what are the critical points of f? 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 $x = \square$
(simplify your answer. use a comma to separate answers as needed)
○ b. the function f has no critical points.

Explanation:

Step1: Find where \(f^{\prime}(x) = 0\)

Set \(f^{\prime}(x)=\frac{x^{2}(x - 2)}{x + 5}=0\). Since a fraction is zero when the numerator is zero (and the denominator is non - zero), we solve \(x^{2}(x - 2)=0\).
Using the zero - product property \(a\times b = 0\Rightarrow a = 0\) or \(b = 0\). For \(x^{2}(x - 2)=0\), we have \(x^{2}=0\) gives \(x = 0\) and \(x-2=0\) gives \(x = 2\).

Step2: Find where \(f^{\prime}(x)\) is undefined

The function \(y = f^{\prime}(x)\) is undefined when the denominator \(x + 5=0\), so \(x=-5\). But critical points are in the domain of \(f(x)\). If \(x=-5\) is not in the domain of \(f(x)\) (since \(x
eq - 5\) for \(f^{\prime}(x)\)), we don't consider it as a critical point.

Answer:

A. The critical point(s) of \(f\) is/are \(x = 0,2\)