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if-x-y-z-gt-0-and-xyz-27-prove-that-1-x-2-27-1-y-2-27-1-z-2-27-1-12-




Question Number 158508 by HongKing last updated on 05/Nov/21
if  x;y;z>0  and  xyz=27  prove that:  (1/(x^2 +27)) + (1/(y^2 +27)) + (1/(z^2 +27)) ≤ (1/(12))
$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{xyz}=\mathrm{27}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} +\mathrm{27}}\:+\:\frac{\mathrm{1}}{\mathrm{y}^{\mathrm{2}} +\mathrm{27}}\:+\:\frac{\mathrm{1}}{\mathrm{z}^{\mathrm{2}} +\mathrm{27}}\:\leqslant\:\frac{\mathrm{1}}{\mathrm{12}} \\ $$

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