# What are the critical points of f(x) =ln(xe^x)?

Oct 23, 2017

The function is defined only for $x > 0$ and using the properties of logarithms:

$\ln \left(x {e}^{x}\right) = \ln x + \ln \left({e}^{x}\right) = x + \ln x$

Both functions are monotone increasing so their sum has no local minimum or maximums and in fact:

$\frac{d}{\mathrm{dx}} \ln \left(x {e}^{x}\right) = 1 + \frac{1}{x} \ne 0$ for $x > 0$

graph{ln(xe^x) [-10, 10, -5, 5]}