Question

answer the following questions about the function whose derivative is ( f^{prime}(x)=(x - 7)^{2}(x + 9) ).
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: Recall the definition of critical points

Critical points of a function \(y = f(x)\) are the values of \(x\) where \(f^{\prime}(x)=0\) or \(f^{\prime}(x)\) is undefined. Since \(f^{\prime}(x)=(x - 7)^{2}(x + 9)\) is a polynomial, it is defined for all real \(x\).

Step2: Set \(f^{\prime}(x)=0\)

Set \((x - 7)^{2}(x + 9)=0\). Using the zero - product property \(ab = 0\) implies \(a = 0\) or \(b=0\).
If \((x - 7)^{2}=0\), then \(x-7 = 0\), so \(x = 7\).
If \(x + 9=0\), then \(x=-9\).

Answer:

The critical point(s) of \(f\) is/are \(x=-9,7\)