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Question Number 116798 by mnjuly1970 last updated on 06/Oct/20

               ...  calculus...         a,b,c ∈R^(+ ) ::            find        min((√( ((b+c)/a))) +(√((a+c)/b)) +(√((a+b)/c)) )=???               ... m.n.1970...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:\:{calculus}... \\ $$$$ \\ $$$$\:\:\:\:\:{a},{b},{c}\:\in\mathbb{R}^{+\:} :: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{find} \\ $$$$ \\ $$$$\:\:\:\:{min}\left(\sqrt{\:\frac{{b}+{c}}{{a}}}\:+\sqrt{\frac{{a}+{c}}{{b}}}\:+\sqrt{\frac{{a}+{b}}{{c}}}\:\right)=??? \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\:\:\:...\:{m}.{n}.\mathrm{1970}... \\ $$$$\:\: \\ $$

Answered by mr W last updated on 06/Oct/20

≥(√((2(√(bc)))/a))+(√((2(√(ac)))/b))+(√((2(√(ab)))/c))  ≥3((√((8abc)/(abc))))^(1/3) =3(√2)

$$\geqslant\sqrt{\frac{\mathrm{2}\sqrt{{bc}}}{{a}}}+\sqrt{\frac{\mathrm{2}\sqrt{{ac}}}{{b}}}+\sqrt{\frac{\mathrm{2}\sqrt{{ab}}}{{c}}} \\ $$$$\geqslant\mathrm{3}\sqrt[{\mathrm{3}}]{\sqrt{\frac{\mathrm{8}{abc}}{{abc}}}}=\mathrm{3}\sqrt{\mathrm{2}} \\ $$

Commented by mnjuly1970 last updated on 07/Oct/20

tayyeballah  thank you..

$${tayyeballah}\:\:{thank}\:{you}.. \\ $$

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