this is similar to section 3.7 problem 18: qianbuscalc2 3.7.018.r. for the function y = 4xe^4x, determine…

Explanation:

Step1: Find the derivative

Use the product - rule $(uv)^\prime = u^\prime v+uv^\prime$, where $u = 4x$ and $v=e^{4x}$. $u^\prime=4$ and $v^\prime = 4e^{4x}$. Then $y^\prime=4e^{4x}+4x\cdot4e^{4x}=4e^{4x}(1 + 4x)$.

Step2: Find critical points

Set $y^\prime = 0$. Since $e^{4x}>0$ for all real $x$, then $1 + 4x=0$, which gives $x=-\frac{1}{4}$. But $x =-\frac{1}{4}\notin[0,2]$, so we only consider the endpoints of the interval $x = 0$ and $x = 2$.

Step3: Evaluate the function at endpoints

When $x = 0$, $y=4\times0\times e^{4\times0}=0$. When $x = 2$, $y=4\times2\times e^{4\times2}=8e^{8}\approx8\times2980.958 = 23847.664\approx23847.7$.

Answer:

Absolute maximum = $23847.7$ at $x = 2$. Absolute minimum = $0$ at $x = 0$.