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- Feb 14, 2012

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

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Remember to read the POTW submission guidelines to find out how to submit your answers!

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- Thread starter
- Admin
- #1

- Feb 14, 2012

- 3,894

-----

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

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!

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- #2

- Feb 14, 2012

- 3,894

Therefore any activities involving me in this forum will be delayed until I feel much better.

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- Feb 14, 2012

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Unfortunately, no one answered last two week's POTW. You can read the suggested solution of other as follows:

$M=\sqrt{y-x}+\sqrt{z-y}+\sqrt{z-x}$

Since $\sqrt{y-x}+\sqrt{z-y}\le \sqrt{2[(y-x)+(z-y)]}=\sqrt{2(z-x)}$, we have

$M\le \sqrt{2(z-x)}+\sqrt{z-x}=(\sqrt{2}+1)\sqrt{z-x}\le \sqrt{2}+1$

The equality holds if and only if $y-x=z-y,\,x=0, z=1,\,y=\dfrac{1}{2}$.

Hence, $\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}\le \sqrt{2}+1$.

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