Question

For what values of x is `f(x)= xe^-x` concave or convex?

Answer 1

`f(x) = xe^(-x)` is concave in `(-oo,2)` and convex in `(2,+oo)` having an inflection point in `x=2`

Explanation:

We need to solve the inequality:

`f''(x) > 0`

so we start by calculating the second derivative of the function:

`f(x) = xe^(-x)`

using the product rule:

`f'(x) = e^(-x) -xe^(-x) = e^(-x)(1-x)`

and again:

`f''(x) = -e^(-x)-e^(-x)(1-x) = e^(-x)(x-2)`

Now to solve the inequality we need to consider that:

`e^(-x) > 0` for every `x`, so:

`f''(x) >0`

`e^(-x)(x-2) > 0`

`(x-2) > 0`

`x > 2`

Thus `f(x)` is concave in `(-oo,2)` and convex in `(2,+oo)` having an inflection point in `x=2`

graph{xe^(-x) [-10, 10, -5, 5]}