# relations between Hessian matrix and local extrema

Let $x$ be a vector, and let $H(x)$ be the Hessian^{} for $f$ at a point $x$. Let $f$ have continuous^{} partial derivatives^{}
of first and second order in a neighborhood^{} of $x$. Let $\nabla f(x)=0$.

If $H(x)$ is positive definite^{} (http://planetmath.org/PositiveDefinite), then $x$ is a strict local minimum for $f$.

If $x$ is a local minimum for $x$, then $H(x)$ is positive semidefinite.

If $H(x)$ is negative definite (http://planetmath.org/NegativeDefinite), then $x$ is a strict local maximum for $f$.

If $x$ is a local maximum for $x$, then $H(x)$ is negative semidefinite.

If $H(x)$ is indefinite, $x$ is a nondegenerate saddle point.

If the case when the dimension^{} of $x$ is 1 (i.e. $f:\mathbb{R}\to \mathbb{R}$), this reduces to the Second Derivative Test^{}, which is as follows:

Let the neighborhood of $x$ be in the domain for $f$, and let $f$ have continuous partial derivatives of first and second order. Let ${f}^{\prime}(x)=0$. If ${f}^{\prime \prime}(x)>0$, then $x$ is a strict local minimum. If $$, then $x$ is a strict local maximum. In the case that ${f}^{\prime \prime}(x)=0$, being ${f}^{\prime \prime \prime}(x)\ne 0$, $x$ is said to be an inflexion point (also called turning point). A typical example is $f(x)=\mathrm{sin}x$, ${f}^{\prime \prime}(x)=-\mathrm{sin}x=0$, $x=n\pi $, $n=0,\pm 1,\pm 2,\mathrm{\dots}$, ${f}^{\prime \prime \prime}(x)=-\mathrm{cos}x$, ${f}^{\prime \prime \prime}(n\pi )=-\mathrm{cos}n\pi ={(-1)}^{n+1}\ne 0$.

Title | relations between Hessian matrix and local extrema |
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Canonical name | RelationsBetweenHessianMatrixAndLocalExtrema |

Date of creation | 2013-03-22 12:59:52 |

Last modified on | 2013-03-22 12:59:52 |

Owner | bshanks (153) |

Last modified by | bshanks (153) |

Numerical id | 14 |

Author | bshanks (153) |

Entry type | Result |

Classification | msc 26B12 |

Related topic | Extrema |

Related topic | Extremum |

Related topic | HessianForm |

Related topic | TestsForLocalExtremaForLagrangeMultiplierMethod |

Defines | second derivative test |