this is similar to section 3.4 problem 8: for the function f(x)=2xe^{3x}, determine its critical number(s)…

this is similar to section 3.4 problem 8: for the function f(x)=2xe^{3x}, determine its critical number(s) if exists. use fractions. use \dne\ if the function has no critical number. answer: x= hint: follow example 1. symbolic formatting help

this is similar to section 3.4 problem 8: for the function f(x)=2xe^{3x}, determine its critical number(s) if exists. use fractions. use \dne\ if the function has no critical number. answer: x= hint: follow example 1. symbolic formatting help

Answer

Explanation:

Step1: Find the derivative

Use the product - rule $(uv)^\prime = u^\prime v+uv^\prime$, where $u = 2x$ and $v = e^{3x}$. The derivative of $u = 2x$ is $u^\prime=2$, and the derivative of $v = e^{3x}$ is $v^\prime = 3e^{3x}$. Then $f^\prime(x)=2e^{3x}+2x\times3e^{3x}=2e^{3x}(1 + 3x)$.

Step2: Set the derivative equal to zero

Set $f^\prime(x)=0$. Since $e^{3x}>0$ for all real - valued $x$, we solve $1 + 3x=0$. $1+3x = 0$ implies $3x=-1$.

Step3: Solve for x

$x=-\frac{1}{3}$.

Answer:

$x =-\frac{1}{3}$