Question

answer the questions below about the function whose derivative is ( f^{prime}(x)=\frac{(x - 2)(x + 8)}{(x + 1)(x - 3)}, x
eq - 1,3 ).
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=-8,2 ) (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 \((x - 2)(x + 8)=0\), so \(x = 2\) or \(x=-8\). \(x=-1\) and \(x = 3\) make \(f^{\prime}(x)\) undefined but are not in the domain of \(f(x)\) (since \(f^{\prime}(x)\) is the derivative). So critical points are \(x=-8,2\).

Step2: Determine sign of \(f^{\prime}(x)\)

Create intervals \((-\infty,-8)\), \((-8,-1)\), \((-1,2)\), \((2,3)\), \((3,\infty)\).

• For \(x\in(-\infty,-8)\), test \(x=-9\): \(f^{\prime}(-9)=\frac{(-9 - 2)(-9 + 8)}{(-9+1)(-9 - 3)}=\frac{(-11)(-1)}{(-8)(-12)}=\frac{11}{96}>0\).
• For \(x\in(-8,-1)\), test \(x=-2\): \(f^{\prime}(-2)=\frac{(-2 - 2)(-2 + 8)}{(-2+1)(-2 - 3)}=\frac{(-4)(6)}{(-1)(-5)}=-\frac{24}{5}<0\).
• For \(x\in(-1,2)\), test \(x = 0\): \(f^{\prime}(0)=\frac{(0 - 2)(0 + 8)}{(0+1)(0 - 3)}=\frac{(-2)(8)}{(1)(-3)}=\frac{16}{3}>0\).
• For \(x\in(2,3)\), test \(x=\frac{5}{2}\): \(f^{\prime}(\frac{5}{2})=\frac{(\frac{5}{2}-2)(\frac{5}{2}+8)}{(\frac{5}{2}+1)(\frac{5}{2}-3)}=\frac{(\frac{1}{2})(\frac{21}{2})}{(\frac{7}{2})(-\frac{1}{2})}=-\frac{21}{7}=-3<0\).
• For \(x\in(3,\infty)\), test \(x = 4\): \(f^{\prime}(4)=\frac{(4 - 2)(4 + 8)}{(4+1)(4 - 3)}=\frac{(2)(12)}{(5)(1)}=\frac{24}{5}>0\).

Answer:

a. \(x=-8,2\)
b. The function \(f\) is increasing on the intervals \((-\infty,-8)\cup(-1,2)\cup(3,\infty)\)