The derivative f'(x) is undefined when x ≤ 0, but since we're only looking at the function f for x > 0, we don't find any critical points because f'(x) is undefined. To find where f'(x) = 0, we find where the numerator is equal to 0.
lnx - 1 = 0
lnx = 1,
x = e.
This is our only critical point. This makes sense on the graph, if we zoom in a lot: