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Question Number 176174 by Matica last updated on 14/Sep/22

a,b,c ∈R_(+ ) ^∗ and abc=1. Prove ((c (√(a^3 +b^3 )))/(a^2 +b^2 )) + ((b (√(a^3 +c^3 )))/(a^2 +c^2 )) + ((a(√(b^3 +c^3 )))/(b^2 +c^2 )) ≥ (3/( (√2)))

$$ \\ $$$$\:\mathrm{a},{b},{c}\:\in\mathbb{R}_{+\:} ^{\ast} \:\:\mathrm{and}\:{abc}=\mathrm{1}.\:\mathrm{Prove} \\ $$$$\:\:\:\frac{{c}\:\sqrt{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} }}{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\:+\:\frac{{b}\:\sqrt{{a}^{\mathrm{3}} +{c}^{\mathrm{3}} }}{{a}^{\mathrm{2}} +{c}^{\mathrm{2}} }\:+\:\frac{{a}\sqrt{{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }}{{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }\:\geqslant\:\frac{\mathrm{3}}{\:\sqrt{\mathrm{2}}} \\ $$$$ \\ $$

Answered by behi834171 last updated on 14/Sep/22

Σ((c(√(a^3 +b^3 )))/(a^2 +b^2 ))≥3(((abc.Π(a^3 +b^3 ))/(Π(a^2 +b^2 ))))^(1/3) ≥ ≥3(((2(√(a^3 b^3 ))×2(√(b^3 c^3 ))×2(√(c^3 a^3 )))/(2(√(a^2 b^2 ))×2(√(b^2 c^2 ))×2(√(c^2 a^2 )))))^(1/3) ≥ ≥3((abc))^(1/3) =3≥(3/( (√2))) .■

$$\Sigma\frac{{c}\sqrt{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} }}{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\geqslant\mathrm{3}\sqrt[{\mathrm{3}}]{\frac{{abc}.\Pi\left({a}^{\mathrm{3}} +{b}^{\mathrm{3}} \right)}{\Pi\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)}}\geqslant \\ $$$$\geqslant\mathrm{3}\sqrt[{\mathrm{3}}]{\frac{\mathrm{2}\sqrt{{a}^{\mathrm{3}} {b}^{\mathrm{3}} }×\mathrm{2}\sqrt{{b}^{\mathrm{3}} {c}^{\mathrm{3}} }×\mathrm{2}\sqrt{{c}^{\mathrm{3}} {a}^{\mathrm{3}} }}{\mathrm{2}\sqrt{{a}^{\mathrm{2}} {b}^{\mathrm{2}} }×\mathrm{2}\sqrt{{b}^{\mathrm{2}} {c}^{\mathrm{2}} }×\mathrm{2}\sqrt{{c}^{\mathrm{2}} {a}^{\mathrm{2}} }}}\geqslant \\ $$$$\geqslant\mathrm{3}\sqrt[{\mathrm{3}}]{{abc}}=\mathrm{3}\geqslant\frac{\mathrm{3}}{\:\sqrt{\mathrm{2}}}\:\:.\blacksquare \\ $$

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