Let $x,y,z\in[0,1]$. Find the maximum value of $\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}$.

Question

Let $x,y,z\in[0,1]$. Then find the maximum value of $\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}$.

Now the given answer is $2\sqrt{2}$ but I am not able to obtain the corresponding values of $x,y,z$. Though when I take $|x-y|=|y-z|=\frac{1}{2}$ and $|x-z|=1$, I do obtain $1+\sqrt{2}$. Can there be a proper method to get to the answer.

Answer 1

WLOG fix $x\geq y\geq z$. Then our expression $E$ is $$E=\sqrt{x-y}+\sqrt{y-z}+\sqrt{x-z}$$ Now, note that as $x$ increases, so does $E$ and so for the maximum value of $E$ we should have $x=1$. Similarly, as $z$ increases, $E$ decreases so we want $z=0$. Then, the expression reduces to $$E=1+\sqrt{y}+\sqrt{1-y}$$ which I hope you can take from here.

Hope this helps. :)