if x;y>0 then: 10 ∙ (√((x^2 +y^2 )/2)) + ((8xy)/(x+y)) ≥ 7x+7y


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Question Number 144917 by mathdanisur last updated on 30/Jun/21

$i f x; y > 0 t h e n :$ $10 \cdot \sqrt{\frac{x^{2} + y^{2}}{2}} + \frac{8 x y}{x + y} ⩾ 7 x + 7 y$
	Answered by justtry last updated on 30/Jun/21

	$r e m e m b e r :$ $Q M ⩾ A M$ $\sqrt{\frac{x^{2} + y^{2}}{2}} ⩾ \frac{x + y}{2} . . . . . (i)$ $A M ⩾ H M$ $\frac{x + y}{2} ⩾ \frac{2}{\frac{1}{x} + \frac{1}{y}} = \frac{2 x y}{x + y}$ $\Leftrightarrow x + y ⩾ \frac{4 x y}{x + y}$ $\Leftrightarrow 2 (x + y) ⩾ \frac{8 x y}{x + y} . . . . . (i i)$ $10 . \sqrt{\frac{x^{2} + y^{2}}{2}} + \frac{8 x y}{x + y} ⩾ 10 . \frac{x + y}{2} + 2 x + 2 y$ $⩾ 5 x + 5 y + 2 x + 2 y$ $⩾ 7 x + 7 y$ $10 \sqrt{\frac{x^{2} + y^{2}}{2}} + \frac{8 x y}{x + y} ⩾ 7 x + 7 y$
	Commented bymathdanisur last updated on 30/Jun/21

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