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

for 0 < x

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*.