Answer

We use the quotient rule to find the derivative:

This is defined for all x in the domain of f, that is, all x greater than zero. It's zero when

ln x = 1

which means when

x = e.

Since the denominator of the derivative is (ln x)², which is always positive, the sign of f'(x) is determined by the sign of the numerator

ln x - 1.

This is negative when x < e and positive when x > e, so we find this numberline:

Therefore f (x) is decreasing to x = e and then increasing, so f (x) hits a minimum at x = e.