If a, b and c are positive real numbers, prove that 4(a³ + b³) ≥ (a + b)³

**Mozart** · October 1st 2009, 04:50 AM

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

**bram kierkels** · October 1st 2009, 05:58 AM

Originally Posted by Mozart

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;
$4a^3+4b^3\geq a^3+3a^2b+3ab^2+b^3$
$\Longleftrightarrow 3a^3+3b^3\geq 3a^2b+3ab^2$
$\Longleftrightarrow a^3+b^3\geq a^2b+ab^2$
$\Longleftrightarrow (a+b)(a^2-ab+b^2)\geq (a+b)(ab)$
$\Longleftrightarrow a^2-ab+b^2\geq ab$
$\Longleftrightarrow a^2-2ab+b^2\geq 0$
$\Longleftrightarrow (a-b)^2\geq 0$

**Krizalid** · October 1st 2009, 11:58 AM

that's wrong proof.

to fix it, in each step you need to add an $\iff.$

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