<s>[INST] Prove That $ \sqrt{\frac xy + \frac yz}+ \sqrt{\frac yz + \frac zx}+ \sqrt{\frac zx + \frac xy}\ge 3 $ for x, y, z $\gt 0$, prove that: $$ \sqrt{\frac xy + \frac yz}+ \sqrt{\frac yz + \frac zx}+ \sqrt{\frac zx + \frac xy}\ge 3 $$
 [/INST] Also, we can use Minkowski and AM-GM
$$\sum_{cyc}\sqrt{\frac{x}{y}+\frac{y}{z}}\geq\sqrt{\left(\sum_{cyc}\sqrt{\frac{x}{y}}\right)^2+\left(\sum_{cyc}\sqrt{\frac{y}{z}}\right)^2}\geq\sqrt{3^2+3^2}=3\sqrt2\geq3.$$
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