Question

Analyzing a Graph In Exercise, analyze and sketch the graph of the function. Label any relative extrema, points of inflection, and asymptotes.

f(x) = x3ex

Answer 1

Sketched & Analyzed.

Explanation:

We are given:
`f(x)=x^3e^x`

First derivative of f(x):

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

There are a root at x = -3 and double roots at x = 0.

So it is decreasing on (-∞, -3) and increasing on (-3, ∞).

Second derivative of f(x):

`f''(x)=e^xx^3+6e^xx^2+6e^xx=xe^x(x^2+6x+6)`

Roots are
`x = 0,\: x = -3-\sqrt3, \:x=-3+\sqrt3`

So it is concave down on
`(-\infty,-3-\sqrt3)` ∪
`(-3-\sqrt3,0)` and concave up on
`(-3-\sqrt3,-3+\sqrt3)` ∪
`(0,\infty)`

The sketch is given in the attachment.