<s>[INST] Given the positive real numbers $0\le a,b,c\le 2$ and $a+b+c=3$. Prove that $a^3+b^3+c^3\le 9$ Given the positive real numbers $$0\le a,b,c\le 2$$ and $$a+b+c=3$$. Prove that $$a^3+b^3+c^3\le 9$$
 [/INST] Let $a\geq b\geq c$.
Since $f(x)=x^3$ is a convex function on $[0,3]$ and $(2,1,0)\succ(a,b,c)$,
by Karamata we obtain $$9=2^3+1^3+0^2\geq a^3+b^3+c^3$$
and we are done!
</s>