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# Critical Point

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A point is called a critical point of a continuously differentiable function if

grad t 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 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.