Saddle point  In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function formula_1 has a critical point at formula_2 that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the formula_3-direction. The name derives from the fact that the prototypical example in two dimensions is a surface that "curves up" in one direction, and "curves down" in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle. In terms of contour lines, a saddle point in two dimensions gives rise to a contour graph or trace in which the contour corresponding to the saddle point's value appears to intersect itself. A simple criterion for checking if a given stationary point of a real-valued function "F"("x","y") of two real variables is a saddle point is to compute the function's Hessian matrix at that point: if the Hessian is indefinite, then that point is a saddle point. For example, the Hessian matrix of the function formula_4 at the stationary point formula_5 is the matrix which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point formula_7 is a saddle point for the function formula_8 but the Hessian matrix of this function at the origin is the null matrix, which is not indefinite. In the most general terms, a saddle point for a smooth function (whose graph is a curve, surface or hypersurface) is a stationary point such that the curve/surface/etc. in the neighborhood of that point is not entirely on any side of the tangent space at that point. In a domain of one dimension, a saddle point is a point which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum. A saddle surface is a smooth surface containing one or more saddle points.