Understanding the second derivative test

Question

Let $f(x,y)$ be a function of class $C^2$. The second derivative test for local extremum seems to rely on the fact that if $f(x,y)$ is stationary at $(0,0)$, then locally $$f(x,y)-f(0,0)\approx f_{xx}(0,0)x^2+2f_{xy}(0,0)xy+f_{yy}(0,0)y^2,$$ or, more precisely, $$f(x,y)-f(0,0)\approx f_{xx}(cx,cy)x^2+2f_{xy}(cx,cy)xy+f_{yy}(cx,cy)y^2.$$ for some $c\in(0,1)$, coming from the error term in Taylor's theorem. What I'm having trouble with is the following: a function $f(x,y)$ can have a very "wavy" saddle at $(0,0)$ (like a ruff, with many foldings) while a quadratic polynomial can only have an "ordinary-looking" saddle. It seems strange that every saddle point of $f(x,y)$ looks locally like an ordinary saddle. Am I misunderstanding something?

Answer 1

Not at all. The idea here is that near a critical point the surface looks like a quadric. “Near enough” might be have to be quite close indeed if the function is very “wavy,” just as with the first derivative you might have to stay quite close to a point for the tangent plane to be a good approximation.