Let's determine the $x$ for which the

$$
|13-x|+|58-x|+|71-x|+|75-x|+|79-x|
$$

sum is minimal!
We show that $S$ has a minimum at $x=71$. Here, the value of $S$ is 83, and if $x>71$, then

$$
S \geq(x-13)+(x-58)+(x-71)+(75-x)+(79-x)=(x-71)+83>83
$$

since for any number $a$, $|a| \geq a$ and $|a| \geq -a$ are both true. Similarly, we find that if $x<71$, then

$$
S \geq(13-x)+(58-x)+(71-x)+(x-75)+(x-79)=(71-x)+83>83
$$

Remark. For any numbers $a_{1} \leq a_{2} \leq \ldots \leq a_{n}$, the sum

$$
S=\left|a_{1}-x\right|+\left|a_{2}-x\right|+\ldots+\left|a_{n}-x\right|
$$

has a minimum at $x=a_{k}$ if $n=2 k-1$, and if $n=2 k$, the value of $S$ is constant between $a_{k}$ and $a_{k+1}$, and is greater everywhere else.