Find all critical numbers and use the second derivitive test to determine all local extrema?

Question

f(x) = xe^-x

Answer 1

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

Checking f''(x)
xe^-x - 2e^-x = 0
e^-x(x - 2) = 0
x = 2 for change of inflection
for x < 2 inflection is negative, so slope is decreasing, so the critical point is a maximum.