# How do you find the critical numbers for f(x) = x-2ln(x) to determine the maximum and minimum?

Aug 13, 2017

The first derivative is given by

$f ' \left(x\right) = 1 - \frac{2}{x}$

Which has critical points at $0$ and when $f ' \left(x\right) = 0$.

$0 = 1 - \frac{2}{x} \to \frac{2}{x} = 1 \to x = 2$

However, $x = 0$ is not really a critical point because the initial function is undefined there. Recall that $\ln \left(0\right) = \emptyset$. Now let's see if $x = 2$ is a maximum or a minimum. At $x = 1$, the function is decreasing because $f ' \left(1\right) < 0$. Hence, $x = 2$ will be an absolute minimum.

Hopefully this helps!