answer the questions below about the function whose derivative is ( f^{prime}(x)=\frac{(x - 5)(x + 9)}{(x +…

answer the questions below about the function whose derivative is ( f^{prime}(x)=\frac{(x - 5)(x + 9)}{(x + 3)(x - 7)}, x\neq-3,7 ).\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.\na. ( x=square ) (use comma to separate answers as needed)\nb. the function ( f ) has no critical points.

answer the questions below about the function whose derivative is ( f^{prime}(x)=\frac{(x - 5)(x + 9)}{(x + 3)(x - 7)}, x\neq-3,7 ).\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.\na. ( x=square ) (use comma to separate answers as needed)\nb. 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)=\frac{(x - 5)(x + 9)}{(x + 3)(x - 7)}=0). Using the zero - product property (a\times b = 0) if (a = 0) or (b=0), we have (x-5 = 0) or (x + 9=0). Solving (x-5 = 0) gives (x = 5), and solving (x + 9=0) gives (x=-9). The derivative (f^{\prime}(x)) is undefined at (x=-3) and (x = 7), but these are not in the domain of the function (since the original function's derivative has a denominator ((x + 3)(x - 7)) and we consider the domain of (f(x)) where (f^{\prime}(x)) is well - defined for the purpose of critical points related to the function's behavior).

Answer:

(x=-9,5)