# Fermat’s theorem (stationary points)

Let $f:(a,b)\to \mathbb{R}$ be a continuous function^{} and suppose that
${x}_{0}\in (a,b)$ is a local extremum of $f$. If $f$ is differentiable^{} in ${x}_{0}$ then ${f}^{\prime}({x}_{0})=0$.

Moreover if $f$ has a local maximum^{} at $a$ and $f$ is differentiable at $a$ (the right derivative exists)
then ${f}^{\prime}(a)\le 0$; if $f$ has a local minimum at $a$ then ${f}^{\prime}(a)\ge 0$.
If $f$ is differentiable in $b$ and
has a local maximum at $b$ then ${f}^{\prime}(b)\ge 0$ while if it has a local minimum at $b$ then ${f}^{\prime}(b)\le 0$.

Title | Fermat’s theorem (stationary points) |
---|---|

Canonical name | FermatsTheoremstationaryPoints |

Date of creation | 2013-03-22 13:45:05 |

Last modified on | 2013-03-22 13:45:05 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 7 |

Author | paolini (1187) |

Entry type | Theorem |

Classification | msc 26A06 |

Related topic | ProofOfLeastAndReatestValueOfFunction |

Related topic | LeastAndGreatestValueOfFunction |