Real numbers $b>a>0$ are given. Find the number $r$ in $[a,b]$ which minimizes the value of $\max\left\{\left|\frac{r-x}x\right||a\le x\le b\right\}$.
Source: 2001 Moldova MO Grade 9 P1
Tags: inequalities, optimization
Real numbers $b>a>0$ are given. Find the number $r$ in $[a,b]$ which minimizes the value of $\max\left\{\left|\frac{r-x}x\right||a\le x\le b\right\}$.