If a, b and c are positive real numbers, prove that 4(a³ + b³) ≥ (a + b)³ & that 9(a³ + b³ + c³) ≥ (a + b + c)³
for the first one;
$\displaystyle 4a^3+4b^3\geq a^3+3a^2b+3ab^2+b^3$
$\displaystyle \Longleftrightarrow 3a^3+3b^3\geq 3a^2b+3ab^2$
$\displaystyle \Longleftrightarrow a^3+b^3\geq a^2b+ab^2$
$\displaystyle \Longleftrightarrow (a+b)(a^2-ab+b^2)\geq (a+b)(ab)$
$\displaystyle \Longleftrightarrow a^2-ab+b^2\geq ab $
$\displaystyle \Longleftrightarrow a^2-2ab+b^2\geq 0$
$\displaystyle \Longleftrightarrow (a-b)^2\geq 0$