<s>[INST] Use the AGM inequality to find the maximum Use the AGM inequality to find the maximum of $(5+\sqrt{x^4+1}) \cdot (9-\sqrt{x^4 + 1})$.
$$ab \le (\frac{a+b}{2})^2$$
I don't know how to relate this inequality to find the maximum of that.
 [/INST] Take $a=5+\sqrt{x^4+1}, b=9-\sqrt{x^4+1}$. Then:
$$ab\le 49$$
Equality occurs when $x=\pm\sqrt[4]{3}$.
Because:
$$ab=(5+\sqrt{x^4+1})(9-\sqrt{x^4+1})\le \left(\frac{(5+\sqrt{x^4+1})+(9-\sqrt{x^4+1})}{2}\right)^2=7^2=49.$$
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