Question

answer the questions below about the function whose derivative is $f(x)=\frac{(x - 5)(x + 9)}{(x + 3)(x - 7)},x\
eq-3,7$.
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. $x=-9,5$ (use comma to separate answers as needed)
b. the function f has no critical points.
b. on what open intervals is f increasing? select the correct choice below and, if necessary, fill in the answer box within your choice
a. the function f is increasing on the interval(s)
(type your answer in interval notation. use a comma to separate answers as needed )
b. the function f is not increasing anywhere.

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 \(a\times b = 0\) implies \(a = 0\) or \(b=0\), so \(x-5 = 0\) gives \(x = 5\) and \(x + 9=0\) gives \(x=-9\). The derivative is undefined at \(x=-3\) and \(x = 7\), but these are not in the domain of the original function (since the derivative is a rational function and the original function's domain excludes \(x=-3\) and \(x = 7\)). So the critical points are \(x=-9\) and \(x = 5\).

Step2: Test intervals for increasing/decreasing

We use the critical points \(x=-9\) and \(x = 5\) to divide the number line into intervals: \((-\infty,-9)\), \((-9,5)\), \((5,\infty)\).

• For the interval \((-\infty,-9)\), let \(x=-10\). Then \(f^{\prime}(-10)=\frac{(-10 - 5)(-10 + 9)}{(-10+3)(-10 - 7)}=\frac{(-15)(-1)}{(-7)(-17)}=\frac{15}{119}>0\).
• For the interval \((-9,5)\), let \(x = 0\). Then \(f^{\prime}(0)=\frac{(0 - 5)(0 + 9)}{(0+3)(0 - 7)}=\frac{(-5)(9)}{(3)(-7)}=\frac{-45}{-21}=\frac{15}{7}>0\).
• For the interval \((5,\infty)\), let \(x=6\). Then \(f^{\prime}(6)=\frac{(6 - 5)(6 + 9)}{(6+3)(6 - 7)}=\frac{(1)(15)}{(9)(-1)}=-\frac{5}{3}<0\).

Answer:

a. \(x=-9,5\)
b. The function \(f\) is increasing on the intervals \((-\infty,-9)\) and \((-9,5)\)