AM-GM inequality

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.


am-gm-inequality


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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