answer the questions below about the function whose derivative is ( f^{prime}(x)=\frac{(x - 2)(x + 8)}{(x +…

answer the questions below about the function whose derivative is ( f^{prime}(x)=\frac{(x - 2)(x + 8)}{(x + 1)(x - 3)}, x\neq - 1,3 ).\na. what are the critical points of ( f )?\nb. on what open intervals is ( f ) increasing or decreasing?\nc. at what points, if any, does ( f ) assume local maximum and minimum values?\na. ( x=-8,2 ) (use comma to separate answers as needed)\nb. the function ( f ) has no critical points.\nb. on what open intervals is ( f ) increasing? select the correct choice below and, if necessary, fill in the answer box within your choice.\na. the function ( f ) is increasing on the interval(s) (type your answer in interval notation. use a comma to separate answers as needed)\nb. the function ( f ) is not increasing anywhere.

answer the questions below about the function whose derivative is ( f^{prime}(x)=\frac{(x - 2)(x + 8)}{(x + 1)(x - 3)}, x\neq - 1,3 ).\na. what are the critical points of ( f )?\nb. on what open intervals is ( f ) increasing or decreasing?\nc. at what points, if any, does ( f ) assume local maximum and minimum values?\na. ( x=-8,2 ) (use comma to separate answers as needed)\nb. the function ( f ) has no critical points.\nb. on what open intervals is ( f ) increasing? select the correct choice below and, if necessary, fill in the answer box within your choice.\na. the function ( f ) is increasing on the interval(s) (type your answer in interval notation. use a comma to separate answers as needed)\nb. the function ( f ) is not increasing anywhere.

Answer

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))