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Mathematics-Online lexicon:

Critical Point

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A point $ x$ is called a critical point of a continuously differentiable function $ f$ if

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

If the second partial derivatives of $ f$ are continuous at $ x$ 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 $ f$. A critical point $ x$ is called At an elliptic point, $ f$ 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 $ f$. At a saddle point, there exist directions with increasing and decreasing function values.
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  automatically generated 9/22/2016