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

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Expert Answers

rcmath | 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 =>`

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

`f(-3)=(-3)^3*e^(-3)=-27/(e^3)`

Hence **the abs min **is `(-3,-27/e^3)`

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

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