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)=6xe^(-0.5x). find the location of any inflection points of f(x). select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. there is an inflection point at x= (type an integer or a decimal. use a comma to separate answers as needed.) b. there are no inflection points.

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of f(x)=6xe^(-0.5x). find the location of any inflection points of f(x). select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. there is an inflection point at x= (type an integer or a decimal. use a comma to separate answers as needed.) b. there are no inflection points.

Answer

Explanation:

Step1: Find the first - derivative

Use the product rule $(uv)^\prime = u^\prime v+uv^\prime$, where $u = 6x$ and $v=e^{-0.5x}$. $u^\prime=6$ and $v^\prime=- 0.5e^{-0.5x}$. So $f^\prime(x)=6e^{-0.5x}+6x(-0.5e^{-0.5x})=6e^{-0.5x}(1 - 0.5x)$.

Step2: Find the second - derivative

Use the product rule again on $f^\prime(x)=6e^{-0.5x}(1 - 0.5x)$. Let $u = 6(1 - 0.5x)$ and $v = e^{-0.5x}$. $u^\prime=-3$ and $v^\prime=-0.5e^{-0.5x}$. $f^{\prime\prime}(x)=-3e^{-0.5x}+6(1 - 0.5x)(-0.5e^{-0.5x})=e^{-0.5x}(-3 - 3+1.5x)=e^{-0.5x}(1.5x - 6)$.

Step3: Find inflection points

Set $f^{\prime\prime}(x)=0$. Since $e^{-0.5x}\gt0$ for all real $x$, we solve $1.5x - 6 = 0$. $1.5x=6$, then $x = 4$.

Answer:

A. There is an inflection point at $x = 4$