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# Inequalities

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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

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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