# extended mean-value theorem

Let $f:[a,b]\to \mathbb{R}$ and $g:[a,b]\to \mathbb{R}$ be continuous^{} on $[a,b]$ and differentiable^{} on $(a,b)$. Then there exists some number $\xi \in (a,b)$ satisfying:

$$(f(b)-f(a)){g}^{\prime}(\xi )=(g(b)-g(a)){f}^{\prime}(\xi ).$$ |

If $g$ is linear this becomes the usual mean-value theorem.

Title | extended mean-value theorem |
---|---|

Canonical name | ExtendedMeanvalueTheorem |

Date of creation | 2013-03-22 13:04:11 |

Last modified on | 2013-03-22 13:04:11 |

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 9 |

Author | mathwizard (128) |

Entry type | Theorem |

Classification | msc 26A06 |

Synonym | Cauchy’s mean value theorem |

Synonym | extended mean value theorem |

Synonym | generalized mean value theorem |

Related topic | MeanValueTheorem |