# AM-GM inequality

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

The *arithmetic mean* of numbers is .

The *geometric mean* of numbers is .

The AM-GM Inequality states that:

For any nonnegative numbers ,

, and equality holds if and only if .

## How to Apply?

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

Yes! Using the AM-GM inequality we have just learnt above, we know .

Cubing both sides, we have, .

Also, the AM-GM inequality tells us that there is equality only when , i.e. . Hence, the largest possible product is 1000.

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