<s>[INST] If $a,b,c>0$ and $abc=1\;,$ Then minimum value of Expression. 
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
 [/INST] 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.
</s>