Find first and second derivatives, critical number, inflection points.

f(x) = xe^x

f'(x) = ('x) (e^x) + x(e^x)('x)

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

f'(x) = e^x(1+x)

Critical number x = -1

f"(x) = e^x(1) + (1+x)(e^x)

f"(x) = e^x +e^x(1+x)

Critical Number x=-1

I hope I am close to the correct answer.