<s>[INST] Proving $4(a^3 + b^3) \ge (a + b)^3$ and $9(a^3 + b^3 + c^3) \ge (a + b + c)^3$ 
Let $a$, $b$ and $c$ be positive real numbers.
  $(\mathrm{i})$ Prove that $4(a^3 + b^3) \ge (a + b)^3$.
  $(\mathrm{ii})$Prove that $9(a^3 + b^3 + c^3) \ge (a + b + c)^3.$

For the first one I tried expanding to get $a^3 + b^3 \ge a^2b+ab^2$ but I'm not sure how to prove it.
 [/INST] for (i);
notice that $(a^3+b^3)=(a+b)(a^2-ab+b^2)$.  This is the standard factorization for the sum of cubes.  Then $4(a^2-ab+b^2)\ge(a+b)^2$.
Thus, 
$$4a^2-4ab+4b^2\ge (a+b)^2$$
$$a^2+2ab+b^2+3a^2-6ab+3b^2\ge (a+b)^2$$
$$(a+b)^2+3(a-b)^2\ge(a+b)^2$$
Since $3(a-b)^2$ is always positive or $0$, the inequality is proven.
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