<s>[INST] Is it inequality true Please help me prove the inequality:
$$\sqrt[3]{3+\sqrt[3] 3}+\sqrt[3]{3-\sqrt[3] 3}<2\sqrt[3] 3$$
Thanky for your help and your attention.
 [/INST] We can use $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)$.
Since $a^2+b^2+c^2-ab-ac-bc=0\Leftrightarrow a=b=c$
and $\sqrt[3]{3+\sqrt[3]3}\neq\sqrt[3]{3-\sqrt[3]3}$ , we need to prove that
$$3+\sqrt[3]3+3-\sqrt[3]3-8\cdot3+3\cdot2\sqrt[3]3\sqrt[3]{9-\sqrt[3]9}<0$$ or
$$\sqrt[3]{27-3\sqrt[3]9}<3$$
which is obvious.
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