What is the least value of $f(x) = x - 2\log x$ on $[1, e]$?

Explanation

1. 01

   $f'(x) = 1 - \frac{2}{x} = 0$ at $x = 2$, which lies in $[1, e]$.

2. 02

   $f(1) = 1$, $f(2) = 2 - 2\log 2 \approx 0.61$, $f(e) = e - 2 \approx 0.72$. The least is $2 - 2\log 2$.