[home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] | ||

Mathematics-Online lexicon: | ||

## Extrema of Multivariate Functions |

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z | overview |

Let be a scalar function of variables. Suppose that has continuous second partial derivatives.

A point is called a minimum point for and is called a local minimum of if there is a - ball (of positive radius) centered at such that

for all

A point is called a maximum point for and is called a local maximum of if there is a - ball (of positive radius) centered at such that

for all

(An - ball is in the case a disk and for just an ordinary ball.)

is called a local extremum of if it is either a minimum or maximum.

If is a local minimum (maximum) of of , then

grad

A sufficient condition for a local minimum (maximum) is that all eigenvalues of the Hesse matrix at are positive (negative).

If there are eigenvalues with different signs, then is a saddle point ( is a hyperbolic critical point). If at least one eigenvalue is zero and all eigenvalues different from zero have the same sign (i.e is a parabolic critical point), then it is impossible to decide only with the second partial derivatives whether is a local extremum.

()

**Annotation:**

- Beweis: Lokales Extremum (german)

automatically generated 8/20/2008 |