answer the questions below about the function whose derivative is $f(x)=\\frac{(x - 2)(x + 8)}{(x + 1)(x…

answer the questions below about the function whose derivative is $f(x)=\\frac{(x - 2)(x + 8)}{(x + 1)(x - 3)},x\\neq-1,3$.\n\n a. what are the critical points of f?\n b. on what open intervals is f increasing or decreasing?\n c. at what points, if any, does f assume local maximum and minimum values?\n\n a. the function f is increasing on the interval(s) $(-\\infty,-8),(-1,2),(3,\\infty)$\n (type your answer in interval notation. use a comma to separate answers as needed )\n b. the function f is not increasing anywhere.\n on what open intervals is f decreasing? select the correct choice below and, if necessary, fill in the answer box within your choice.\n a. the function f is decreasing on the interval(s) \n (type your answer in interval notation. use a comma to separate answers as needed )\n b. the function f is not decreasing anywhere.

answer the questions below about the function whose derivative is $f(x)=\\frac{(x - 2)(x + 8)}{(x + 1)(x - 3)},x\\neq-1,3$.\n\n a. what are the critical points of f?\n b. on what open intervals is f increasing or decreasing?\n c. at what points, if any, does f assume local maximum and minimum values?\n\n a. the function f is increasing on the interval(s) $(-\\infty,-8),(-1,2),(3,\\infty)$\n (type your answer in interval notation. use a comma to separate answers as needed )\n b. the function f is not increasing anywhere.\n on what open intervals is f decreasing? select the correct choice below and, if necessary, fill in the answer box within your choice.\n a. the function f is decreasing on the interval(s) \n (type your answer in interval notation. use a comma to separate answers as needed )\n b. the function f is not decreasing anywhere.

Answer

Explanation:

Step1: Analyze the sign of (f^{\prime}(x))

We have (f^{\prime}(x)=\frac{(x - 2)(x + 8)}{(x + 1)(x - 3)}). The critical points (where (f^{\prime}(x)=0) or is undefined) are (x=-8,x=-1,x = 2,x=3). These points divide the real - line into intervals ((-\infty,-8),(-8,-1),(-1,2),(2,3),(3,\infty)). Take test points:

  • For the interval ((-\infty,-8)), let (x=-9). Then (f^{\prime}(-9)=\frac{(-9 - 2)(-9 + 8)}{(-9 + 1)(-9 - 3)}=\frac{(-11)(-1)}{(-8)(-12)}=\frac{11}{96}>0).
  • For the interval ((-8,-1)), let (x=-2). Then (f^{\prime}(-2)=\frac{(-2 - 2)(-2 + 8)}{(-2 + 1)(-2 - 3)}=\frac{(-4)(6)}{(-1)(-5)}=\frac{-24}{5}<0).
  • For the interval ((-1,2)), let (x = 0). Then (f^{\prime}(0)=\frac{(0 - 2)(0 + 8)}{(0 + 1)(0 - 3)}=\frac{(-2)(8)}{(1)(-3)}=\frac{16}{3}>0).
  • For the interval ((2,3)), let (x=\frac{5}{2}). Then (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 the interval ((3,\infty)), let (x = 4). Then (f^{\prime}(4)=\frac{(4 - 2)(4 + 8)}{(4 + 1)(4 - 3)}=\frac{(2)(12)}{(5)(1)}=\frac{24}{5}>0).

Step2: Determine the decreasing intervals

Since (f^{\prime}(x)<0) on the intervals ((-8,-1)) and ((2,3)).

Answer:

A. The function (f) is decreasing on the interval(s) ((-8,-1),(2,3))