I'm having trouble understanding a proof in a book I'm reading. It's not really homework because I'm reading it on my own time, but it seems more appropriate to post here than the general math forum.(adsbygoogle = window.adsbygoogle || []).push({});

The exercise is to show that:

[tex]x+y+z \leq 2\left\{\frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y}\right\}[/tex]

The book says that to prove it, "We apply Cauchy's inequality to the splitting:

[tex]x+y+z=\frac{x}{\sqrt{y+z}}\sqrt{y+z}+\frac{y}{\sqrt{x+z}}\sqrt{x+z}+\frac{z}{\sqrt{x+y}}\sqrt{x+y}[/tex]"

First, I've never seen set brackets {} used like this. Is it the same as normal brackets in this context (multiply what's inside the brackets by 2)?

And I really don't understand how to prove the first statement with the second... am I supposed to apply Cauchy's inequality to every element of the right hand of the second equation?

If I do that, after simplifying, I get:

[tex]x+y+z\leq\sqrt{\frac{x^2}{y+z}}\sqrt{y+z}+\sqrt{\frac{y^2}{x+z}}\sqrt{x+z}+\sqrt{\frac{z^2}{x+y}}\sqrt{x+y}[/tex]

Which looks like nothing to me... it just goes back to x+y+z=x+y+z, which doesn't help me much.

I'm really stuck here :grumpy: please help. Thanks!

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# Homework Help: Trouble Understanding Proof Using Cauchy's Inequality

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