Question

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

Answer 1

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

`ln(xe^x) = lnx + ln(e^x) = x+lnx`

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

`d/dx ln(xe^x) = 1+1/x !=0` for `x>0`

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