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1. Saddle Point

   Linked via "second partial derivatives"

   $$\nabla f(\mathbf{x}_0) = \mathbf{0}$$
   The nature of this critical point is determined by the Hessian matrix ($\mathbf{H}(\mathbf{x}_0)$), which contains the second partial derivatives. The eigenvalues of the Hessian matrix dictate the local curvature of the function.
   A critical point $\mathbf{x}0$ is classified as a saddle point if and only if the Hessian matrix $\mathbf{H}(\mathbf{x}0)$ has both positive and negative [eigenvalues](/entries/…