Question Number 145530 by mathdanisur last updated on 05/Jul/21
$$\left(\frac{\mathrm{1}}{\mathrm{2}}\:+\:\frac{\mid{a}\mid\:+\:\mid{b}\mid}{\mid{a}\:+\:{b}\mid}\right)_{\boldsymbol{{min}}} =\:? \\ $$
Answered by puissant last updated on 07/Jul/21
$$\mid\mathrm{a}+\mathrm{b}\mid\leqslant\mid\mathrm{a}\mid+\mid\mathrm{b}\mid\:\Rightarrow\:\frac{\mid\mathrm{a}\mid+\mid\mathrm{b}\mid}{\mid\mathrm{a}+\mathrm{b}\mid}\:\geqslant\mathrm{1} \\ $$$$\Rightarrow\:\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mid\mathrm{a}\mid+\mid\mathrm{b}\mid}{\mid\mathrm{a}+\mathrm{b}\mid}\right)\geqslant\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\mathrm{take}\:\mathrm{min}=\frac{\mathrm{3}}{\mathrm{2}}.. \\ $$
Commented by mathdanisur last updated on 07/Jul/21
$${Thanks}\:{Ser} \\ $$