AM-GM inequality

If x and y are two non-negative numbers, then

A.M. =
G.M. =
We have A.M. G.M.
i.e,
This is known as AM-GM inequality.
Proof:
We have
(x – y)2 0
x2 – 2xy + y2 0
x2 + 2xy + y2 – 4xy 0
(x + y)2 – 4xy 0
(x + y)2 4xy
x + y 2
A.M. G.M.