<s>[INST] Prove inequality $a^3b^3+2b^3c^3+3a^3c^3\le 0$ For $a^3+b^3+c^3=0$. Prove that $a^3b^3+2b^3c^3+3a^3c^3\le 0$

i think from $a^3+b^3+c^3=0$ we have one of the three numbers is zero, the other two are opposites.Suppose $a;b$ opposites and $c=0$
 [/INST] $$a^3b^3+2b^3c^3+3a^3c^3=a^3b^3-(a^3+b^3)(3a^3+2b^3)=$$
$$=-(3a^6+4a^3b^3+2b^6)=-a^6-2(a^3+b^3)^2\leq0.$$
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