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Extrema of Multivariate Functions

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

A point $ x_*$ is called a minimum point for $ f$ and $ f(x_*)$ is called a local minimum of $ f$ if there is a $ n$ - ball $ B$ (of positive radius) centered at $ x_*$ such that

$\displaystyle f(x) \geq f(x_*) \ $   for all$\displaystyle \ x \in B .$

A point $ x_*$ is called a maximum point for $ f$ and $ f(x_*)$ is called a local maximum of $ f$ if there is a $ n$ - ball $ B$ (of positive radius) centered at $ x_*$ such that

$\displaystyle f(x) \leq f(x_*) \ $   for all$\displaystyle \ x \in B .$

(An $ n$ - ball is in the case $ n=2$ a disk and for $ n=3$ just an ordinary ball.)

$ f(x_*)$ is called a local extremum of $ f$ if it is either a minimum or maximum.

If $ f(x_*)$ is a local minimum (maximum) of $ f$ of $ x_*$ , then

   grad$\displaystyle \,f(x_*) = 0\, . $

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

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

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  automatically generated 8/20/2008