Lesson AM-GM inequality

This Lesson (AM-GM inequality ) was created by by Shin123(626)

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It is a fact that for any 2 nonnegative numbers, the arithmetic mean is always greater than or equal to the geometric mean of the 2 numbers.

Proof

The arithmetic mean of 2 numbers x and y is %28x%2By%29%2F2

and the geometric mean is sqrt%28xy%29

So we are trying to prove that
%28x%2By%29%2F2%3E=sqrt%28xy%29

Square both sides to get
%28x%5E2%2B2xy%2By%5E2%29%2F4%3E=xy

Move all terms to one side.
%28%28x%5E2%2B2xy%2By%5E2%29%2F4%29-xy%3E=0

Find common denominator.
%28%28x%5E2%2B2xy%2By%5E2%29%2F4%29-4xy%2F4%3E=0

Multiply both sides by 4.
x%5E2-2xy%2By%5E2%3E=0

Factor left side.
%28x-y%29%5E2%3E=0

Since the square of any real number is always greater than or equal to 0, the proof is done.

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