# AM-GM inequality

A very useful inequality in Mathematics is the AM-GM Inequality.

The arithmetic mean of numbers $x_1, x_2, \cdots, x_n$ is $\displaystyle \boxed{\frac{x_1+ x_2+\cdots+x_n}{n}}$.

The geometric mean of numbers $x_1, x_2, \cdots, x_n$ is $\boxed{\sqrt[n]{x_1\cdot x_2 \cdots x_n}}$.

The AM-GM Inequality states that:

For any nonnegative numbers $x_1, x_2, \cdots, x_n$,

$\displaystyle\boxed{\frac{x_1+x_2+\cdots+ x_n}{n}\geq\sqrt[n]{x_1\cdot x_2 \cdots x_n}}$, and equality holds if and only if $x_1=x_2=\cdots=x_n$.

## How to Apply?

Let say we have three (nonnegative) numbers a, b, c that add up to 30, i.e. $a+b+c=30$. Can we know what is the largest possible product $abc$?

Yes! Using the AM-GM inequality we have just learnt above, we know $\displaystyle \frac{a+b+c}{3}\geq \sqrt[3]{abc}$.

$\displaystyle 10\geq \sqrt[3]{abc}$

Cubing both sides, we have, $\displaystyle abc\leq 10^3=1000$.

Also, the AM-GM inequality tells us that there is equality only when $a=b=c$, i.e. $a=b=c=10$. Hence, the largest possible product $abc$ is 1000.

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