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

answer the following questions about the function whose derivative is $f(x)=\\frac{x^{2}(x - 2)}{x + 5},x\\neq-5$.\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 or minimum values?\nb. on what open intervals is f increasing or decreasing? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\n○ a. the function f is increasing on the open interval(s) and never decreasing\n(type your answer in interval notation. use a comma to separate answers as needed.)\n○ b. the function f is increasing on the open interval(s) $(2,\\infty)$ and decreasing on the open interval(s)\n$(-\\infty,2)$\n(type your answers in interval notation. use a comma to separate answers as needed.)\n○ c. the function f is decreasing on the open interval(s) and never increasing\n(type your answer in interval notation. use a comma to separate answers as needed.)

answer the following questions about the function whose derivative is $f(x)=\\frac{x^{2}(x - 2)}{x + 5},x\\neq-5$.\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 or minimum values?\nb. on what open intervals is f increasing or decreasing? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\n○ a. the function f is increasing on the open interval(s) and never decreasing\n(type your answer in interval notation. use a comma to separate answers as needed.)\n○ b. the function f is increasing on the open interval(s) $(2,\\infty)$ and decreasing on the open interval(s)\n$(-\\infty,2)$\n(type your answers in interval notation. use a comma to separate answers as needed.)\n○ c. the function f is decreasing on the open interval(s) and never increasing\n(type your answer in interval notation. use a comma to separate answers as needed.)

Answer

Explanation:

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

We have (f^{\prime}(x)=\frac{x^{2}(x - 2)}{x + 5}), (x\neq-5). First, find the critical points from (f^{\prime}(x)=0) (i.e., (x^{2}(x - 2)=0)) which gives (x = 0) or (x=2), and the point where (f^{\prime}(x)) is undefined (x=-5). Now, consider the intervals ((-\infty,-5)), ((-5,0)), ((0,2)) and ((2,\infty)). Take a test - point in each interval:

  • For the interval ((-\infty,-5)), let (x=-6). Then (f^{\prime}(-6)=\frac{(-6)^{2}(-6 - 2)}{-6 + 5}=\frac{36\times(-8)}{-1}=288>0).
  • For the interval ((-5,0)), let (x=-1). Then (f^{\prime}(-1)=\frac{(-1)^{2}(-1 - 2)}{-1 + 5}=\frac{1\times(-3)}{4}=-\frac{3}{4}<0).
  • For the interval ((0,2)), let (x = 1). Then (f^{\prime}(1)=\frac{1^{2}(1 - 2)}{1+5}=\frac{1\times(-1)}{6}=-\frac{1}{6}<0).
  • For the interval ((2,\infty)), let (x = 3). Then (f^{\prime}(3)=\frac{3^{2}(3 - 2)}{3 + 5}=\frac{9\times1}{8}=\frac{9}{8}>0).

Step2: Determine increasing and decreasing intervals

Since (f^{\prime}(x)>0) on ((-\infty,-5)\cup(2,\infty)) and (f^{\prime}(x)<0) on ((-5,2)) (but (x\neq - 5)). The function (y = f(x)) is increasing when (f^{\prime}(x)>0) and decreasing when (f^{\prime}(x)<0).

Answer:

The function (f) is increasing on the open intervals ((-\infty,-5)) and ((2,\infty)) and decreasing on the open interval ((-5,2)).