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Consider the function $f(x,y)=2xy-x^3-y^2$. One of the stationary points is $(0,0)$. At this point, $f_{xx}f_{yy}-f_{xy}f_{yx}<0$. According to me, this indicates that (0,0) is a saddle point. However, the text I am referring to calls this "neither an extremum nor a saddle point". Am I missing something?

Edit The plot (from GeoGebra) looks like this:

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4 Answers

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You're right, and there's a mistake in the example. I'm pretty sure something like $x^3+y^2$ was intended; that's genuinely not a saddle point, despite increasing in some directions and decreasing in others.

This is also dependent on the definition; some sources define a saddle point to be a critical point that's not a maximum or minimum, in which case this situation would be impossible.

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If you follow the path $y=x$ then $f(x,x)=x^2-x^3$ meaning a local minimum. If you follow $y=-x$ then $f(x,-x)=-3x^2-x^3$ meaning a local maximum. These behaviors match $g(x,y)=xy$, an archetypal saddle point at the origin.

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here's what the function looks like (would have posted as a comment but can't)

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From graphical plot, it appears to be a saddle point having positive and negative curvature along mutually perpendicular directions

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