<s>[INST] How do we know that $x^2 + \frac{1}{x^2}$ is greater or equal to $2$? For one problem, we were supposed to know that: 
$$x^2 + \frac{1}{x^2}\geq 2.$$
How do you deduce this instantly when looking at the expression above?
 [/INST] $x^2 + \frac{1}{x^2} = \frac{x^4 + 1}{x^2}$
$     = \frac{x^4 - 2x^2 + 1}{x^2} + 2$
$     = \frac{(x^2 - 1)^2}{x^2} + 2$
$     \geq 0 + 2 = 2$
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