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 $f(x)=\\frac{(x - 2)(x + 8)}{(x + 1)(x - 3)},x\\neq-1,3$.\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. what are the critical points of f? select the correct choice below and, if necessary, fill in the answer box within your choice.\n○ a. $x=\\square$ (use comma to separate answers as needed)\n○ b. the function f has no critical points.

answer the questions below about the function whose derivative is $f(x)=\\frac{(x - 2)(x + 8)}{(x + 1)(x - 3)},x\\neq-1,3$.\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. what are the critical points of f? select the correct choice below and, if necessary, fill in the answer box within your choice.\n○ a. $x=\\square$ (use comma to separate answers as needed)\n○ b. the function f has no critical points.

Answer

Explanation:

Step1: Find critical points

Critical points occur where (f^{\prime}(x) = 0) or (f^{\prime}(x)) is undefined. Set (f^{\prime}(x)=0): (\frac{(x - 2)(x + 8)}{(x + 1)(x - 3)}=0). A fraction is zero when the numerator is zero (and denominator is non - zero). So, ((x - 2)(x + 8)=0). Using the zero - product property (x-2 = 0) gives (x = 2) and (x+8=0) gives (x=-8). The derivative (f^{\prime}(x)) is undefined when the denominator ((x + 1)(x - 3)=0), i.e., (x=-1) or (x = 3). But critical points are in the domain of (f(x)). Assuming (f(x)) is differentiable (since we are given (f^{\prime}(x)) in the form of a rational function for (x\neq - 1,3)), the critical points are (x=-8) and (x = 2).

Answer:

A. (x=-8,2)