<s>[INST] Range Of Quartic Polynomial Of Two Variables $a$,$b$ are real numbers such that $~3\leq a^{2}+ab+b^{2}\leq 6$.    
I would like to find the range of $~a^{4}+b^{4}$.
Is it possible to find it with (well-known) AM-GM, CS, etc...?
 [/INST] $ab \le \dfrac{a^2+b^2}{2} \implies \dfrac{3(a^2+b^2)}{2} \ge 3 \implies a^2+b^2 \ge2 \implies a^4+b^4 \ge 2(\dfrac{a^2+b^2}{2})^2 \ge 2 $  when $a=b=1$ hold "="
$a^4+b^4=(a^2+b^2+\sqrt{2}ab)(a^2+b^2-\sqrt{2}ab)=(u+(\sqrt{2}-1)ab)(u-(\sqrt{2}+1)ab)=u^2-2abu-a^2b^2, u=a^2+ab+b^2 \le 6$
it hints that when $ab<0$, the max will be got. let $c=-b>0, 3\le a^2-ac+c^2\le6  \implies ac \le6 $   when $a=c=\sqrt{6}$ hold "="
$a^4+b^4 \le 36+12ac-(ac)^2=72-(6-ac)^2\le 72$,when $a=c=-b=\sqrt{6}$ get "="
so  we get : $2\le a^4+b^4 \le 72$
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