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if-x-y-gt-0-peove-that-2-3-2-1-x-1-y-1-3-x-2-y-2-2-xy-




Question Number 151383 by mathdanisur last updated on 20/Aug/21
if  x;y>0  peove that:  (2/3) ∙ (2/((1/x)+(1/y))) + (1/3) ∙ (√((x^2 +y^2 )/2)) ≥ xy
$$\mathrm{if}\:\:\mathrm{x};\mathrm{y}>\mathrm{0}\:\:\mathrm{peove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{2}}{\mathrm{3}}\:\centerdot\:\frac{\mathrm{2}}{\frac{\mathrm{1}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{y}}}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\:\centerdot\:\sqrt{\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }{\mathrm{2}}}\:\geqslant\:\mathrm{xy} \\ $$
Commented by mathdanisur last updated on 20/Aug/21
Thank You Dear Ser, ≥((xy))^(1/3)  for solution please
$$\mathrm{Thank}\:\mathrm{You}\:\mathrm{Dear}\:\boldsymbol{\mathrm{S}}\mathrm{er},\:\geqslant\sqrt[{\mathrm{3}}]{\mathrm{xy}}\:\mathrm{for}\:\mathrm{solution}\:\mathrm{please} \\ $$
Answered by dumitrel last updated on 20/Aug/21
Commented by mathdanisur last updated on 20/Aug/21
Nice Dear Ser, Thank You
$$\mathrm{Nice}\:\mathrm{Dear}\:\boldsymbol{\mathrm{S}}\mathrm{er},\:\mathrm{Thank}\:\mathrm{You} \\ $$

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