Applying Product Rule

[JSmith]

Given the function

.
a) Determine the intervals of increase and decrease.
b) Determine the absolute minimum value of f(x).

So obviously I have to find the derivative and calculate the critical numbers, then test them and so on....

But what I am stuck on is applying the product rule to the function e^x. I know its derivative is itself, so:

f(x)=x³e^x
f'(x)=(x³)(e^x)+(e^x)(3x²)

then how do I simplify?

 

[MarkFL]

You want to factor so that you may apply the zero-factor property. What is the greatest factor you see common to both terms?

 

[JSmith]

You want to factor so that you may apply the zero-factor property. What is the greatest factor you see common to both terms?

would it be x²​? or e^x??? the greatest factor common to all four terms, or the product of the two sets of terms?

 

[MarkFL]

Your derivative has two terms, and each term consists of factors:

\(\displaystyle f'(x)=x^3e^x+3x^2e^x\)

What is the largest factor common to both terms?