Mathematics-Online lexicon: Critical Point

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	Mathematics-Online lexicon:
	Critical Point

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overview

A point

is called a critical point of a continuously differentiable function

grad $\displaystyle \,f(x)=(0,\ldots,0)^$ t $\displaystyle \,.$

If the second partial derivatives of

are continuous at

as well, the type of the critical point can be classified in terms of the eigenvalues of the Hesse matrix $\left(\partial_\nu\partial_\mu f\right)_{\nu,\mu=1}^n$ of

. A critical point

is called

elliptic, if all eigenvalues are different from zero and have the same sign;
hyperbolic or a saddle point, if there exist eigenvalues with different signs;
parabolic, if at least one eigenvalue is zero and all nonzero eigenvalues have the same sign.

At an elliptic point,

has a local extremum. This might also be the case for a parabolic point. However, a decision requires more information, since higher order partial derivatives can influence the local behavior of

. At a saddle point, there exist directions with increasing and decreasing function values.

see also:

Keyword: Extreme point: function of several variables

[Examples]

automatically generated 9/22/2016