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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