Question

answer the questions below about the function whose derivative is ( f^{prime}(x)=\frac{(x - 5)(x + 9)}{(x + 3)(x - 7)}, x
eq-3,7 ).
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. 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 \(f^{\prime}(x)=\frac{(x - 5)(x + 9)}{(x + 3)(x - 7)}=0\). Using the zero - product property \(a\times b = 0\) if \(a = 0\) or \(b=0\), we have \(x-5 = 0\) or \(x + 9=0\).
Solving \(x-5 = 0\) gives \(x = 5\), and solving \(x + 9=0\) gives \(x=-9\).
The derivative \(f^{\prime}(x)\) is undefined at \(x=-3\) and \(x = 7\), but these are not in the domain of the function (since the original function's derivative has a denominator \((x + 3)(x - 7)\) and we consider the domain of \(f(x)\) where \(f^{\prime}(x)\) is well - defined for the purpose of critical points related to the function's behavior).

Answer:

\(x=-9,5\)