For the function, use the First Derivative Test to determine if each critical point is a minimum, a maximum, or neither.
for 0 < x
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.