Question

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

Answer 1

f'(x) = e^-x + x*-1*e^-x = e^-x -xe^-x
f''(x) = -e^-x - ( e^-x -xe^-x)
f''(x) = -2e^-x + xe^-x
f''(x) = 0
e^-x(-2+x) = 0
2 = x

f''(0) = -2 <- concave down
f''(4) = -2/e^4 + 4/e^4 = 2e^4 <- concave up

so x > 2
(2,infinity)