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. |
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Proof: |
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We have | ||

(x – y)^{2} |
≥ | 0 |

x^{2} – 2xy + y^{2} |
≥ | 0 |

x^{2} + 2xy + y^{2} – 4xy |
≥ | 0 |

(x + y)^{2} – 4xy |
≥ | 0 |

(x + y)^{2} |
≥ | 4xy |

x + y | ≥ | 2 |

≥ | ||

A.M. | ≥ | G.M. |