For what intervals is f(x)=xe^-x concave down?

When the second derivative is negative the function is concave downward.

f(x) = xe^-x

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

f '' (x) = - e^-x - [e^-x - xe^-x] = -e^-x - e^-x + xe^-x = -2e^-x + xe^-x

f ''(x) = e^-x [x - 2]

Found x for f ''(x) < 0

e^-x [x -2] < 0

Given that e^-x is always > 0, x - 2 < 0

=> x < 2

Therefore, the function is concave downward in (- ∞ , 2)