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Question Number 151496 by puissant last updated on 21/Aug/21

∀x,y∈R_+ ^∗ , show that (x/(x^4 +y^2 ))+(y/(y^4 +x^2 ))≤(1/(xy))

$$\forall{x},{y}\in\mathbb{R}_{+} ^{\ast} ,\:{show}\:{that}\:\frac{{x}}{{x}^{\mathrm{4}} +{y}^{\mathrm{2}} }+\frac{{y}}{{y}^{\mathrm{4}} +{x}^{\mathrm{2}} }\leqslant\frac{\mathrm{1}}{{xy}} \\ $$

Answered by dumitrel last updated on 21/Aug/21

(x/(x^4 +y^2 ))+(y/(x^2 +y^4 ))≤(x/(2x^2 y))+(y/(2y^2 x))=(1/(2xy))+(1/(2xy))=(1/(xy))

$$\frac{{x}}{{x}^{\mathrm{4}} +{y}^{\mathrm{2}} }+\frac{{y}}{{x}^{\mathrm{2}} +{y}^{\mathrm{4}} }\leqslant\frac{{x}}{\mathrm{2}{x}^{\mathrm{2}} {y}}+\frac{{y}}{\mathrm{2}{y}^{\mathrm{2}} {x}}=\frac{\mathrm{1}}{\mathrm{2}{xy}}+\frac{\mathrm{1}}{\mathrm{2}{xy}}=\frac{\mathrm{1}}{{xy}} \\ $$

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