PREMISES
In my multivariable calculus book, a saddle point for a function $f: Dom(f) \subseteq \mathbb{R}^n \rightarrow \mathbb{R} $ is defined as a stationary point which is not a minimum or maximum. I will consider the case $n=2$ because it's the case I can actually visualize it (because these functions are 3d surfaces).
It seems to me that this definitionisn't really geometrically specific, since it includes many different "geometric situations". Also, I think that my calculus boook might have given that definition because it talks about stationary points just to then be able to talk about optimizations, and in optimizations you only care about maximum and minimum points. So I found a definition in this paper that seems, to me, more accurate in describing saddle points as I would expect them, so saddle points that recall the (real) saddles.
QUESTION
But then I was interested in points like the ones on the y-axis of $x^3$, which seems a generalization of a one dimensional inflexion point, and in points like $(0,0)$ in $x^3+y^4$ and $x^3-y^4$, which seems to be "characterized" by the fact that going in the $x$ direction "yields" an inflexion point, and going in the $y$ direction "yields" a maximum/minimum.
My question is: are those points "interesting" in any way geometrically or do they have any applications? Would it be useful to classify them? If yes, can someone send me some links where I can find out more about them?
Best Answer
Yes, the general definition of saddle points would include points like the ones you describe (a point $p$ where $\frac{df}{dx} = 0$ at $p$ and $\frac{df}{dx}$ has a local extrema at $p$). It's a catch-all term for points which are neither a minimum or a maximum. So the geometric interpretation is that (a) they have a minimum in one direction and a maximum in another or (b) there is some direction where the derivative is $0$ but doesn't change sign at that point (the classic inflection point, like in $x^3$).
Inflection points like this are "uncommon" in the sense that if you pick a random differentiable one-variable function and look at some particular stationary point of that function $p$, the probability that $p$ is not an extrema of the function is almost zero. So most stationary points that are not extrema do look like saddles, rather than having an inflection point with respect to some variable.
I don't think stationary points are classified into categories finer than maximum, minimum, saddle point. For functions of several variables, you can get some information about the geometry of a stationary point using its Hessian matrix. If you're interested in classification of stationary points I would look into those.