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If-x-y-z-R-and-x-2-y-2-z-2-3-Prove-that-1-4-x-1-4-y-1-4-z-1-

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Question Number 210571 by hardmath last updated on 12/Aug/24
If x,y,z∈R^+ and x^2 +y^2 +z^2 =3 Prove that (1/(4−x)) + (1/(4−y)) + (1/(4−z)) ≤ 1
$$\mathrm{If}\:\:\mathrm{x},\mathrm{y},\mathrm{z}\in\mathrm{R}^{+} \:\:\mathrm{and}\:\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} =\mathrm{3} \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{4}−\mathrm{x}}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{4}−\mathrm{y}}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{4}−\mathrm{z}}\:\:\leqslant\:\:\mathrm{1} \\ $$
Answered by A5T last updated on 15/Aug/24
≡(1/(x−4))+(1/(y−4))+(1/(z−4))≥−1 (1/(x−4))+(1/(y−4))+(1/(z−4))≥(9/(x+y+z−12)) 1=(√((x^2 +y^2 +z^2 )/3))≥((x+y+z)/3)⇒x+y+z≤3 ⇒(9/(x+y+z−12))≥(9/(−9))=−1 ⇒(1/(4−x))+(1/(4−y))+(1/(4−z))≤1
$$\equiv\frac{\mathrm{1}}{{x}−\mathrm{4}}+\frac{\mathrm{1}}{{y}−\mathrm{4}}+\frac{\mathrm{1}}{{z}−\mathrm{4}}\geqslant−\mathrm{1} \\ $$$$\frac{\mathrm{1}}{{x}−\mathrm{4}}+\frac{\mathrm{1}}{{y}−\mathrm{4}}+\frac{\mathrm{1}}{{z}−\mathrm{4}}\geqslant\frac{\mathrm{9}}{{x}+{y}+{z}−\mathrm{12}} \\ $$$$\mathrm{1}=\sqrt{\frac{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} }{\mathrm{3}}}\geqslant\frac{{x}+{y}+{z}}{\mathrm{3}}\Rightarrow{x}+{y}+{z}\leqslant\mathrm{3} \\ $$$$\Rightarrow\frac{\mathrm{9}}{{x}+{y}+{z}−\mathrm{12}}\geqslant\frac{\mathrm{9}}{−\mathrm{9}}=−\mathrm{1} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{4}−{x}}+\frac{\mathrm{1}}{\mathrm{4}−{y}}+\frac{\mathrm{1}}{\mathrm{4}−{z}}\leqslant\mathrm{1} \\ $$

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