<s>[INST] Minimum value of the given function- Find the minimum value of-
$$\frac{(x+\frac{1}{x})^6-(x^6+\frac{1}{x^6})-2}{(x+\frac{1}{x})^3+(x^3+\frac{1}{x^3})}$$
I tried opening the brackets and trying to cancel the terms and using AM-GM.
 [/INST] Write $z=x+\frac1x$. Then the function is equal to
$$\frac{z^6-(z^6-6z^4+9z^2-2)-2}{z^3+z^3-3z}=\frac{6z^4-9z^2}{2z^3-3z}=\frac{3z^2(2z^2-3)}{z(2z^2-3)}$$
Now $z$ can never assume a value in $(-2,2)$ and $2z^2-3=0\implies z=\sqrt{3/2}\in(-2,2)$, so we can cancel and get $3z$. Since $z$ has no minimum over the reals, so does the original function, but over the positive reals it does have a minimum of $6$.
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