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
.
![\displaystyle 10\geq \sqrt[3]{abc}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+10%5Cgeq+%5Csqrt%5B3%5D%7Babc%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
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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