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Question Number 99645 by bramlex last updated on 22/Jun/20

$$Ifx,y>0then\mid \sqrt{xy}-\frac{x+y}{2}\mid +\mid \frac{x+y}{2}+\sqrt{xy}\mid =$$

Commented bybemath last updated on 22/Jun/20

$$\mathrm{since}\sqrt{\mathrm{xy}}>0\mathrm{\&}\frac{\mathrm{x}+\mathrm{y}}{2}>0$$ $$\mathrm{so}\frac{\mathrm{x}+\mathrm{y}}{2}+\sqrt{\mathrm{xy}}>0$$ $$\mathrm{case}\left(1\right)\mathrm{if}\sqrt{\mathrm{xy}}-\frac{\mathrm{x}+\mathrm{y}}{2}>0$$ $$\mathrm{then}\sqrt{\mathrm{xy}}-\frac{\mathrm{x}+\mathrm{y}}{2}+\frac{\mathrm{x}+\mathrm{y}}{2}+\sqrt{\mathrm{xy}}=2\sqrt{\mathrm{xy}}$$ $$\mathrm{case}\left(2\right)\mathrm{if}\sqrt{\mathrm{xy}}-\frac{\mathrm{x}+\mathrm{y}}{2}<0$$ $$\mathrm{then}-\sqrt{\mathrm{xy}}+\frac{\mathrm{x}+\mathrm{y}}{2}+\frac{\mathrm{x}+\mathrm{y}}{2}+\sqrt{\mathrm{xy}}=\mathrm{x}+\mathrm{y}$$ $$$$

Answered by Farruxjano last updated on 22/Jun/20

$$\mathit{W}\mathit{h}\mathit{e}\mathit{n}\mathit{x},\mathit{y}>0\Rightarrow \frac{x+y}{2}+\sqrt{xy}>0\Rightarrow$$ $$\mid \frac{x+y}{2}+\sqrt{xy}\mid =\frac{x+y}{2}+\sqrt{xy}$$ $$\mathit{W}\mathit{e}\mathit{k}\mathit{n}\mathit{o}\mathit{w}\mathit{i}\mathit{n}\mathit{e}\mathit{q}\mathit{u}\mathit{a}\mathit{l}\mathit{i}\mathit{t}\mathit{y}\mathit{K}\mathit{o}\mathit{s}\mathit{h}\mathit{i}:$$ $$\frac{\mathit{x}+\mathit{y}}{2}⩾\sqrt{\mathit{x}\mathit{y}}\mathit{w}\mathit{h}\mathit{e}\mathit{n}\mathit{x},\mathit{y}>0\mathit{S}\mathit{o},$$ $$\mid \sqrt{xy}-\frac{x+y}{2}\mid =\frac{\mathit{x}+\mathit{y}}{2}-\sqrt{\mathit{x}\mathit{y}}$$ $$\mid \sqrt{xy}-\frac{x+y}{2}\mid +\mid \frac{x+y}{2}+\sqrt{xy}\mid =\frac{\mathit{x}+\mathit{y}}{2}-\sqrt{\mathit{x}\mathit{y}}+$$ $$+\frac{x+y}{2}+\sqrt{xy}=\mathit{x}+\mathit{y}▴.$$

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