what is the saddle point
Answers
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.
Answer:
In mathematics, a saddle point or minimax point[1] 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.[2] 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 {\displaystyle f(x,y)=x^{2}+y^{3}}{\displaystyle f(x,y)=x^{2}+y^{3}} has a critical point at {\displaystyle (0,0)}(0,0) 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 {\displaystyle y}y-direction.
A saddle point (in red) on the graph of z=x2−y2 (hyperbolic paraboloid)
Saddle point between two hills (the intersection of the figure-eight {\displaystyle z}z-contour)
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 its