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)