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Find the largest positive real number \(\lambda\) such that for any nonnegative real numbers x, y, and z such that x^2 + y^2 + z^2 = 1, the inequality

 

\(\lambda xy + yz \le \frac{\sqrt{5}}{2}\) holds.

 

Thank you for any help!

 Jun 25, 2021
 #1
avatar+139 
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i asked one person i know about this one, as for me myself am not sure, and he said his first impression was to do $ y(\lambda x +z) \le \sqrt{\frac{5}{2}} $

 

$  y \le\frac{\sqrt{\frac{5}{2}} }{(\lambda x +z)} $

 

any idea what to do next?

 Jun 27, 2021

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