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If $a,b,c>0$ and $abc=1\;,$ Then minimum value of $$\frac{a^2}{a^2+2}+\frac{b^2}{b^2+2}+\frac{c^2}{c^2+2}$$

$\bf{My\; Try::}$ Using $\bf{Cauchy\; Schwarz}$ Inequality

$$\frac{a^2}{a^2+2}+\frac{b^2}{b^2+2}+\frac{c^2}{c^2+2}\geq \frac{(a+b+c)^2}{a^2+b^2+c^2+6}$$

Now How can I solve after that, Help Required

Thanks

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After your manipulation, the only reasonable claim is that: $$(a+b+c)^2 \geq a^2+b^2+c^2+6 $$ that is equivalent to: $$ ab+ac+bc \geq 3 $$ that follows from the AM-GM inequality: $$ ab+ac+bc = 3\cdot AM(ab,ac,bc) \geq 3\cdot GM(ab,ac,bc) = 3.$$ Equality occurs at $a=b=c=1$, as expected.

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Plug in $c = \frac1{ab}$, and set the partial derivatives $\frac{\partial}{\partial a}$ and $\frac{\partial}{\partial b}$ equal to zero to find candidates for a minimum.

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