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. |
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Proof: |
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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. |