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)
