Define saddle points. Compute the saddle points of the function

Question

Define saddle points. Compute the saddle points of the function $f:R^{2} ightarrow R$ given by $f(x,y)=(y-x^{2})(y-2x^{2})$ .

20 Mar 2023

Answer :

Word Count : 407

A saddle point of a function f(x,y) is a point (a,b) such that f(a,b) is a critical point (i.e., the partial derivatives of f with respect to x and y are both zero at (a,b)), and the function changes sign when moving in different directions away from (a,b). Geometrically, this means that the surface defined by f(x,y) has a valley in one direction and a hill in the perpendicular direction at (a,b), like a saddle.

To find the saddle points of the function $f(x,y) = (y-x^2)(y-2x^2)$ , we first compute its partial derivatives:

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