<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] To simplify the notation, replace $\,a^3 \mapsto a, \,b^3 \mapsto b, \,c^3 \mapsto c\,$ so the problem becomes:

for $a+b+c=0\,$, prove that $\,ab+2bc+3ac\le 0$

Then, given that $b+c=-a\,$ and $a+b = -c\,$:
$$
ab+2bc+3ac = (ab+ac)+2(bc+ac)=a(b+c)+2c(a+b)=-a^2 - 2c^2 \le 0
$$
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