<s>[INST] Minimize $\frac{(x^2+1)(y^2+1)(z^2+1)}{ (x+y+z)^2}$, $x,y,z>0$ Minimize $\;\;\displaystyle \frac{(x^2+1)(y^2+1)(z^2+1)}{ (x+y+z)^2}$, if $x,y,z>0$. 
By setting gradient to zero I found $x=y=z=\frac{1}{\displaystyle\sqrt{2}}$, which could minimize the function.

Question from Jalil Hajimir

 [/INST] Let $x=\frac{a}{\sqrt2},$ $y=\frac{b}{\sqrt2}$ and $z=\frac{c}{\sqrt2}.$
Thus, since we can assume that $(a^2-1)(b^2-1)\geq0,$ by C-S we obtain:
$$\frac{(x^2+1)(y^2+1)(z^2+1)}{(x+y+z)^2}=\frac{(a^2+2)(b^2+2)(c^2+2)}{4(a+b+c)^2}\geq$$
$$\geq\frac{3(a^2+b^2+1)(1+1+c^2)}{4(a+b+c)^2}\geq\frac{3(a+b+c)^2}{4(a+b+c)^2}=\frac{3}{4}.$$
The equality occurs for $x=y=z=\frac{1}{\sqrt2},$ which says that we got a minimal value.
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