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)=9e^{-0.3x^{2}} ).\n\nb. the function is increasing on ( (-infty,0) ). it is decreasing on ( (0,infty) ).\n(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.\n(type your answer in interval notation. type integers or decimals. use a comma to separate answers as needed.)\nfind the location of any local extrema of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. there is a local maximum at ( x= ) and there is a local minimum at ( x= ).\n(type integers or decimals. use a comma to separate answers as needed.)\nb. there is a local minimum at ( x= ). there is no local maximum.\n(type an integer or a decimal. use a comma to separate answers as needed.)\nc. there is a local maximum at ( x= ). there is no local minimum.\n(type an integer or a decimal. use a comma to separate answers as needed.)\nd. there are no local extrema.

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( f(x)=9e^{-0.3x^{2}} ).\n\nb. the function is increasing on ( (-infty,0) ). it is decreasing on ( (0,infty) ).\n(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.\n(type your answer in interval notation. type integers or decimals. use a comma to separate answers as needed.)\nfind the location of any local extrema of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. there is a local maximum at ( x= ) and there is a local minimum at ( x= ).\n(type integers or decimals. use a comma to separate answers as needed.)\nb. there is a local minimum at ( x= ). there is no local maximum.\n(type an integer or a decimal. use a comma to separate answers as needed.)\nc. there is a local maximum at ( x= ). there is no local minimum.\n(type an integer or a decimal. use a comma to separate answers as needed.)\nd. there are no local extrema.

Answer

Explanation:

Step1: Find the first derivative

Given (y = 9e^{-0.3x^{2}}), use the chain - rule. If (y = 9e^{u}) and (u=-0.3x^{2}), then (\frac{dy}{du}=9e^{u}) and (\frac{du}{dx}=- 0.6x). By the chain - rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=9e^{-0.3x^{2}}\cdot(-0.6x)=-5.4xe^{-0.3x^{2}}).

Step2: Determine where the function is increasing or decreasing

Set (\frac{dy}{dx}>0) to find where the function is increasing. (-5.4xe^{-0.3x^{2}}>0). Since (e^{-0.3x^{2}}>0) for all (x\in R), then (-5.4x>0), which gives (x < 0). Set (\frac{dy}{dx}<0) to find where the function is decreasing. (-5.4xe^{-0.3x^{2}}<0). Since (e^{-0.3x^{2}}>0) for all (x\in R), then (-5.4x<0), which gives (x>0).

Step3: Find the local extrema

Set (\frac{dy}{dx} = 0), so (-5.4xe^{-0.3x^{2}}=0). Since (e^{-0.3x^{2}}\neq0) for all (x\in R), then (x = 0). Find the second derivative using the product - rule. If (y=-5.4xe^{-0.3x^{2}}), let (u=-5.4x) and (v = e^{-0.3x^{2}}). Then (u^\prime=-5.4) and (v^\prime=-0.6xe^{-0.3x^{2}}). By the product - rule (y^\prime=u^\prime v+uv^\prime=-5.4e^{-0.3x^{2}}+(-5.4x)(-0.6xe^{-0.3x^{2}})=e^{-0.3x^{2}}(-5.4 + 3.24x^{2})). Evaluate the second derivative at (x = 0): (y^{\prime\prime}(0)=e^{0}(-5.4+0)=-5.4<0).

Answer:

For the increasing and decreasing intervals: The function is increasing on ((-\infty,0)) and decreasing on ((0,\infty)). For the local extrema: There is a local maximum at (x = 0). So the answer for the local - extrema part is (C).