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)))


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

$a, b, c \in R_{+}^{*} and a b c = 1 . Prove$ $\frac{c \sqrt{a^{3} + b^{3}}}{a^{2} + b^{2}} + \frac{b \sqrt{a^{3} + c^{3}}}{a^{2} + c^{2}} + \frac{a \sqrt{b^{3} + c^{3}}}{b^{2} + c^{2}} ⩾ \frac{3}{\sqrt{2}}$
	Answered by behi834171 last updated on 14/Sep/22

	$Σ \frac{c \sqrt{a^{3} + b^{3}}}{a^{2} + b^{2}} ⩾ 3 \sqrt[3]{\frac{a b c . Π (a^{3} + b^{3})}{Π (a^{2} + b^{2})}} ⩾$ $⩾ 3 \sqrt[3]{\frac{2 \sqrt{a^{3} b^{3}} \times 2 \sqrt{b^{3} c^{3}} \times 2 \sqrt{c^{3} a^{3}}}{2 \sqrt{a^{2} b^{2}} \times 2 \sqrt{b^{2} c^{2}} \times 2 \sqrt{c^{2} a^{2}}}} ⩾$ $⩾ 3 \sqrt[3]{a b c} = 3 ⩾ \frac{3}{\sqrt{2}} . ◼$
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