Question Number 116798 by mnjuly1970 last updated on 06/Oct/20
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:\:{calculus}... \\ $$$$ \\ $$$$\:\:\:\:\:{a},{b},{c}\:\in\mathbb{R}^{+\:} :: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{find} \\ $$$$ \\ $$$$\:\:\:\:{min}\left(\sqrt{\:\frac{{b}+{c}}{{a}}}\:+\sqrt{\frac{{a}+{c}}{{b}}}\:+\sqrt{\frac{{a}+{b}}{{c}}}\:\right)=??? \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\:\:\:...\:{m}.{n}.\mathrm{1970}... \\ $$$$\:\: \\ $$
Answered by mr W last updated on 06/Oct/20
$$\geqslant\sqrt{\frac{\mathrm{2}\sqrt{{bc}}}{{a}}}+\sqrt{\frac{\mathrm{2}\sqrt{{ac}}}{{b}}}+\sqrt{\frac{\mathrm{2}\sqrt{{ab}}}{{c}}} \\ $$$$\geqslant\mathrm{3}\sqrt[{\mathrm{3}}]{\sqrt{\frac{\mathrm{8}{abc}}{{abc}}}}=\mathrm{3}\sqrt{\mathrm{2}} \\ $$
Commented by mnjuly1970 last updated on 07/Oct/20
$${tayyeballah}\:\:{thank}\:{you}.. \\ $$