Maximize value of the function $A=\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}$

Question

Let $0\le x,y,z\le 2$. Find maximize value of the function $$A=\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}$$

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Outside $(x, y, z)=(0, 1, 2)$ and $A_{\text{max}}=2+\sqrt2$ I have no way in this problem.

Answer 1

WLOG assume $x\le y \le z$. Then we may rewrite the function as $$\sqrt{y-x}+\sqrt{z-y}+\sqrt{z-x}$$ It is now evident it is increasing in $z$ and decreasing in $x$, so we may set $z=2, x=0$, to get $\sqrt y + \sqrt{2-y} +\sqrt2$ to maximise using your favourite method for univariate functions.

I would finish with Jensen ;)