MHB Problem involving arithmetic and geometric mean.

[DrunkenOldFool]

$a,b,c$ are any three positive numbers such that $a+b+c=1$. Prove that

$$ab^2c^3 \leq \frac{1}{432}$$

 

[sbhatnagar]

Consider the 6 numbers

$$a,\frac{b}{2},\frac{b}{2},\frac{c}{3},\frac{c}{3},\frac{c}{3}$$

The arithmetic mean of these numbers is

$\displaystyle AM = \dfrac{a+\frac{b}{2}+\frac{b}{2}+\frac{c}{3}+\frac{c}{3}+\frac{c}{3}}{6}$

$=\frac{1}{6}$

Similarly, you can calculate the Geometric Mean.

$\displaystyle GM=\left( \frac{b}{2}\frac{b}{2}\frac{c}{3}\frac{c}{3}\frac{c}{3}\right)^{\frac{1}{6}}=\left( \frac{ab^2 c^3}{2^2 3^3}\right)^{1 \over 6}$

$AM \geq GM$
$\displaystyle \frac{1}{6} \geq \left( \frac{ab^2 c^3}{2^2 3^3}\right)^{1 \over 6}$

$\displaystyle \Rightarrow \frac{2^23^3}{6^6} \geq ab^2c^3$