summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of (…

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=9 e^{-0.5 x^{2}} ).\na. the function has one vertical asymptote, (type an equation.)\nb. the function has two vertical asymptotes. the leftmost asymptote is and the rightmost asymptote is (type equations.)\nc. there are no vertical asymptotes.\nfind the intervals where ( f(x) ) is increasing or decreasing. select the correct choice below and fill in the answer box(es) to complete your choice.\na. the function is increasing on. it is never decreasing. (type your answer in interval notation. type integers or decimals. use a comma to separate answers as needed.)\nb. the function is increasing on. it is decreasing on (type your answers in interval notation. type integers or decimals. use a comma to separate answers as needed.)\nc. the function is decreasing on. it is never increasing. (type your answer in interval notation. type integers or decimals. use a comma to separate answers as needed.)

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=9 e^{-0.5 x^{2}} ).\na. the function has one vertical asymptote, (type an equation.)\nb. the function has two vertical asymptotes. the leftmost asymptote is and the rightmost asymptote is (type equations.)\nc. there are no vertical asymptotes.\nfind the intervals where ( f(x) ) is increasing or decreasing. select the correct choice below and fill in the answer box(es) to complete your choice.\na. the function is increasing on. it is never decreasing. (type your answer in interval notation. type integers or decimals. use a comma to separate answers as needed.)\nb. the function is increasing on. it is decreasing on (type your answers in interval notation. type integers or decimals. use a comma to separate answers as needed.)\nc. the function is decreasing on. it is never increasing. (type your answer in interval notation. type integers or decimals. use a comma to separate answers as needed.)

Answer

Explanation:

Step1: Find the domain of the function

The function (y = 9e^{-0.5x^{2}}) is an exponential - type function. The domain of (y = e^{u}) where (u=-0.5x^{2}) is all real numbers ((x\in(-\infty,\infty))). Since there are no values of (x) for which the function (y = 9e^{-0.5x^{2}}) is undefined (the exponential function (y = e^{t}) is defined for all (t\in R) and (t=-0.5x^{2}) is a polynomial, which is defined for all (x\in R)), there are no vertical asymptotes.

Step2: Find the first - derivative

Use the chain rule. If (y = 9e^{-0.5x^{2}}), let (u=-0.5x^{2}), then (y = 9e^{u}). The derivative of (y) with respect to (u) is (\frac{dy}{du}=9e^{u}), and the derivative of (u) with respect to (x) is (\frac{du}{dx}=-x). By the chain rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}), so (y^\prime=f^\prime(x)=9e^{-0.5x^{2}}\cdot(-x)=-9xe^{-0.5x^{2}}).

Step3: Find the critical points

Set (y^\prime = 0), so (-9xe^{-0.5x^{2}}=0). Since (e^{-0.5x^{2}}>0) for all (x\in R), then (x = 0) is the critical point.

Step4: Test intervals

  • For the interval ((-\infty,0)), let (x=-1). Then (y^\prime=-9(-1)e^{-0.5(-1)^{2}} = 9e^{-0.5}>0). So the function is increasing on the interval ((-\infty,0)).
  • For the interval ((0,\infty)), let (x = 1). Then (y^\prime=-9(1)e^{-0.5(1)^{2}}=-9e^{-0.5}<0). So the function is decreasing on the interval ((0,\infty)).

Answer:

C. There are no vertical asymptotes. B. The function is increasing on ((-\infty,0)). It is decreasing on ((0,\infty)).