locate the critical points of the following function. then use the second derivative test to determine…

locate the critical points of the following function. then use the second derivative test to determine whether they correspond to local maxima, local minima, or neither. f(x)=e^x(x + 6) what is(are) the critical point(s) of f? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the critical point(s) is(are) x = -7. (use a comma to separate answers as needed.) b. there are no critical points for f. find f(x). f(x)=e^x(x + 8) what is(are) the local minimum/minima of f? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local minimum/minima of f is/are at x = (use a comma to separate answers as needed.) b. there is no local minimum of f.

locate the critical points of the following function. then use the second derivative test to determine whether they correspond to local maxima, local minima, or neither. f(x)=e^x(x + 6) what is(are) the critical point(s) of f? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the critical point(s) is(are) x = -7. (use a comma to separate answers as needed.) b. there are no critical points for f. find f(x). f(x)=e^x(x + 8) what is(are) the local minimum/minima of f? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local minimum/minima of f is/are at x = (use a comma to separate answers as needed.) b. there is no local minimum of f.

Answer

Explanation:

Step1: Recall critical - point definition

Critical points occur where $f^{\prime}(x)=0$ or $f^{\prime}(x)$ is undefined. First, find the first - derivative of $f(x)=e^{x}(x + 6)$ using the product rule $(uv)^\prime=u^\prime v+uv^\prime$, where $u = e^{x}$ and $v=x + 6$. Then $f^{\prime}(x)=e^{x}(x + 6)+e^{x}=e^{x}(x+7)$. Set $f^{\prime}(x)=0$. Since $e^{x}\gt0$ for all $x\in R$, then $x + 7=0$, so $x=-7$.

Step2: Recall second - derivative test

The second - derivative test states that if $f^{\prime}(c)=0$ and $f^{\prime\prime}(c)\gt0$, then $f(x)$ has a local minimum at $x = c$; if $f^{\prime}(c)=0$ and $f^{\prime\prime}(c)\lt0$, then $f(x)$ has a local maximum at $x = c$; if $f^{\prime\prime}(c)=0$, the test is inconclusive. We are given $f^{\prime\prime}(x)=e^{x}(x + 8)$.

Step3: Evaluate $f^{\prime\prime}(x)$ at the critical point

Substitute $x=-7$ into $f^{\prime\prime}(x)$. We get $f^{\prime\prime}(-7)=e^{-7}(-7 + 8)=e^{-7}\gt0$.

Answer:

A. The local minimum/minima of $f$ is/are at $x=-7$