Question

answer the following questions about the function whose derivative is ( f^{prime}(x)=(x - 5)^{2}(x + 7) ).
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.

( \bigcirc ) a. the critical point(s) of ( f ) is/are ( x = ) (simplify your answer. use a comma to separate answers as needed.)

( \bigcirc ) 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 \(f^{\prime}(x)=(x - 5)^{2}(x + 7)\) is a polynomial, it is defined for all \(x\in R\).
Set \(f^{\prime}(x)=0\), then \((x - 5)^{2}(x + 7)=0\).
Using the zero - product property: if \(ab = 0\), then \(a = 0\) or \(b=0\).
For \((x - 5)^{2}=0\), we get \(x = 5\). For \(x + 7=0\), we get \(x=-7\).

Answer:

A. The critical point(s) of \(f\) is/are \(x=-7,5\)