answer the following questions about the function whose derivative is $f(x)=(x - 6)^2(x + 7)$. a. what are…

answer the following questions about the function whose derivative is $f(x)=(x - 6)^2(x + 7)$. 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? (simplify your answer. use a comma to separate answers as needed.) b. the function $f$ has no critical points. b. determine where $f$ is increasing and decreasing. select the correct choice below and fill in the answer box to complete your choice. (type your answer in interval notation. use a comma to separate answers as needed.) a. the function is increasing on the open interval(s) $(-7,6),(6,infty)$, and decreasing on the open interval(s) $(-infty,-7)$. b. the function $f$ is decreasing on the open interval(s) , and never increasing. c. the function $f$ is increasing on the open interval(s) , and never decreasing

answer the following questions about the function whose derivative is $f(x)=(x - 6)^2(x + 7)$. 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? (simplify your answer. use a comma to separate answers as needed.) b. the function $f$ has no critical points. b. determine where $f$ is increasing and decreasing. select the correct choice below and fill in the answer box to complete your choice. (type your answer in interval notation. use a comma to separate answers as needed.) a. the function is increasing on the open interval(s) $(-7,6),(6,infty)$, and decreasing on the open interval(s) $(-infty,-7)$. b. the function $f$ is decreasing on the open interval(s) , and never increasing. c. the function $f$ is increasing on the open interval(s) , and never decreasing

Answer

Explanation:

Step1: Find critical points

Critical points occur where (f^{\prime}(x) = 0). Set ((x - 6)^{2}(x + 7)=0). Using the zero - product property: If ((x - 6)^{2}=0), then (x = 6). If (x+7 = 0), then (x=-7).

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

  • For (x<-7): Let (x=-8), then (f^{\prime}(-8)=(-8 - 6)^{2}(-8 + 7)=(-14)^{2}\times(-1)=196\times(-1)<0).
  • For (-7<x<6): Let (x = 0), then (f^{\prime}(0)=(0 - 6)^{2}(0 + 7)=36\times7>0).
  • For (x>6): Let (x = 7), then (f^{\prime}(7)=(7 - 6)^{2}(7 + 7)=1\times14>0).

Step3: Determine local maxima and minima

Since (f^{\prime}(x)) changes sign from negative to positive at (x=-7), by the first - derivative test, (f(x)) has a local minimum at (x=-7). Since (f^{\prime}(x)) does not change sign at (x = 6) (the sign of (f^{\prime}(x)) is positive on both sides of (x = 6)), (x = 6) is not a local maximum or minimum.

Answer:

a. The critical points of (f) are (x=-7) and (x = 6). b. The function is increasing on the open intervals ((-7,6)\cup(6,\infty)) and decreasing on the open interval ((-\infty,-7)). c. The function (f) has a local minimum at (x=-7) and no local maximum.