Question

find the absolute maximum and minimum values of the following function on the given interval. then graph the function. identify the points on the graph where the absolute extrema occur.
g(x)=3x^{3}e^{-x}, -1≤x≤4
find the absolute maximum. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute maximum value occurs at x=
(use a comma to separate answers as needed. round to three decimal places as needed.)
b. there is no absolute maximum.

Explanation:

Step1: Find the derivative

Use the product - rule $(uv)^\prime = u^\prime v+uv^\prime$, where $u = 3x^{3}$ and $v=e^{-x}$.
$u^\prime=9x^{2}$ and $v^\prime=-e^{-x}$.
So, $g^\prime(x)=9x^{2}e^{-x}-3x^{3}e^{-x}=3x^{2}e^{-x}(3 - x)$.

Step2: Find the critical points

Set $g^\prime(x) = 0$.
Since $3x^{2}e^{-x}(3 - x)=0$, and $e^{-x}>0$ for all real $x$.
$3x^{2}=0$ gives $x = 0$ and $3 - x=0$ gives $x = 3$. Both $x = 0$ and $x = 3$ are in the interval $[-1,4]$.

Step3: Evaluate the function at critical and end - points

Evaluate $g(x)$ at $x=-1,x = 0,x = 3,x = 4$.
$g(-1)=3(-1)^{3}e^{-(-1)}=-3e\approx - 8.155$.
$g(0)=3(0)^{3}e^{-0}=0$.
$g(3)=3(3)^{3}e^{-3}=81e^{-3}\approx3.347$.
$g(4)=3(4)^{3}e^{-4}=192e^{-4}\approx3.547$.

Answer:

A. The absolute maximum value $3.547$ occurs at $x = 4$.