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x-y-z-gt-0-and-x-2-y-2-z-2-3-prove-that-xyz-x-y-z-1-2-2-1-




Question Number 150852 by mathdanisur last updated on 15/Aug/21
x;y;z>0  and  x^2 +y^2 +z^2 =3  prove that  xyz ≤ (((x + y + z - 1)/2))^2  ≤ 1
$$\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} =\mathrm{3}\:\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{xyz}\:\leqslant\:\left(\frac{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\:-\:\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \:\leqslant\:\mathrm{1} \\ $$

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