For the function f(x)=x^(3)e^x, determine intervals of increase and decrease and absolute minimum value of f(x)

Expert Answers
rcmath eNotes educator| Certified Educator

To determine the interval of increase/decrease and the min, we need to find the first derivative.
`f(x)=x^3*e^x =>`

`f'(x)=3x^2*e^x+x^3*e^x =>`


Since the first two factors are always positive the sign of f' depend on 3+x.

`f'(x)>0 => 3+x>0 => x>(-3)`

`f'(x)<0 => 3+x<0 => x<-3`

To find the point of inflections we set f'(x)=0 => x=0 or x=-3

From the graph we can see that x=-3 is the x-coord of the min. (We can also check that by using the 2nd derivative test)


Hence the abs min is `(-3,-27/e^3)`

The function increases over `(-3,oo)`

The function decreases over `(-oo,-3)`