Question

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

Answer 1

The first derivative is given by

`f'(x) = 1 - 2/x`

Which has critical points at `0` and when `f'(x) = 0`.

`0 = 1- 2/x -> 2/x = 1 -> x = 2`

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

Hopefully this helps!