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.


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.

Featured book from Amazon:

Competition Math for Middle School

Written for the gifted math student, the new math coach, the teacher in search of problems and materials to challenge exceptional students, or anyone else interested in advanced mathematical problems. Competition Math contains over 700 examples and problems in the areas of Algebra, Counting, Probability, Number Theory, and Geometry. Examples and full solutions present clear concepts and provide helpful tips and tricks. “I wish I had a book like this when I started my competition career.” Four-Time National Champion MATHCOUNTS coach Jeff Boyd “This book is full of juicy questions and ideas that will enable the reader to excel in MATHCOUNTS and AMC competitions. I recommend it to any students who aspire to be great problem solvers.” Former AHSME Committee Chairman Harold Reiter

Author: mathtuition88

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.