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)=

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)=

Answer

Explanation:

Step1: Find the first - derivative

Use the product rule $(uv)^\prime = u^\prime v+uv^\prime$, where $u = e^{x}$ and $v=x + 6$. $u^\prime=e^{x}$, $v^\prime = 1$. So $f^\prime(x)=e^{x}(x + 6)+e^{x}=e^{x}(x + 7)$. Set $f^\prime(x)=0$, since $e^{x}\gt0$ for all $x\in R$, then $x+7 = 0$, so $x=-7$.

Step2: Find the second - derivative

Differentiate $f^\prime(x)=e^{x}(x + 7)$ using the product rule again. Let $u = e^{x}$, $v=x + 7$, $u^\prime=e^{x}$, $v^\prime = 1$. Then $f^{\prime\prime}(x)=e^{x}(x + 7)+e^{x}=e^{x}(x + 8)$.

Answer:

$f^{\prime\prime}(x)=e^{x}(x + 8)$