We denote the median as $$$m$$$. We have the sorted array $$$a$$$ where some elements are smaller, some are equal and some are larger than $$$m$$$ in this order. Let's consider the other possible choices which are, choosing a smaller value than $$$m$$$ and a larger value than $$$m$$$. If we choose a greater value than $$$m$$$ then the difference between the elements smaller and equal to $$$m$$$ will increase by $$$x - m$$$. And the difference between the elements greater than $$$m$$$ will increase by $$$x - m$$$. But, because there are at least as many elements equal or smaller to $$$m$$$ as elements greater than $$$m$$$ we can see it is not optimal to choose this number, because in the best case, the difference wouldn't change and in the worst case, it would change by $$$x$$$. The same technique can be used for numbers smaller than $$$m$$$. This leaving $$$m$$$ as the best choice.