Problem #PRU-65303

Problem

A numerical set $x_1,\dots ,x_n$ is given. Consider the function $d(t)=\frac{mi{n}_{i=1,\dots ,n}|x_i-t|+ma{x}_{i=1,\dots ,n}|x_i-t|}{2}$.

a) Is it true that the function $d(t)$ takes the smallest value at a single point, for any set of numbers $x_1,\dots ,x_n$?

b) Compare the values of $d(c)$ and $d(m)$ where $c=\frac{mi{n}_{i=1,\dots ,n}{x}_i+ma{x}_{i=1,\dots ,n}{x}_i}{2}$ and $m$ is the median of the specified set.