Question Number 84532 by john santu last updated on 14/Mar/20
$$\mathrm{x}>\:\mathrm{0}\:,\:\mathrm{y}\:>\:\mathrm{0}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{xy}}{\mathrm{x}+\mathrm{y}}\:<\:\mathrm{x} \\ $$
Commented by mr W last updated on 14/Mar/20
$$\Rightarrow{y}<{x}+{y} \\ $$$$\Rightarrow\frac{{y}}{{x}+{y}}<\mathrm{1} \\ $$$$\Rightarrow\frac{{xy}}{{x}+{y}}<{x} \\ $$
Answered by TANMAY PANACEA last updated on 14/Mar/20
$$\frac{{xy}}{{x}+{y}}−{x} \\ $$$$\frac{{xy}−{x}^{\mathrm{2}} −{xy}}{{x}+{y}} \\ $$$$\frac{−{x}^{\mathrm{2}} }{{x}+{y}}<\mathrm{0}\:\:\:\:{so}\:\frac{{xy}}{{x}+{y}}<{x} \\ $$