Find the interval(s) where f(x)=lnx/x is decreasing

Question

Find the interval(s) where f(x)=lnx/x is decreasing

Answer 1

A function is decreasing on an interval I if its first derivative is less than 0 on that interval.

f(x) = lnx / x

We can use the quotient rule to find the first derivative.

f'(x) = [x/x - lnx]/x² < 0

[1 - lnx]/x² < 0

x² is always positive. We can cross multiply it to multiply it by zero and eliminate it.

1 - lnx < 0

-lnx < -1

Multiply by -1 and flip the inequality.

lnx > 1

Exponentiate both sides.

x > e

The function is decreasing on (e, ∞)