I’m currently taking a Calculus class, and we’re learning how to find the critical points and what they are through the second derivative test. Unfortunately, the method that we currently use to figure out WHAT kind of critical point something is doesn’t work on any points where the Hessian (You know, fxxfyy - fxyfyx) is zero, where they are said to be denegrate.
What methods are there of finding points that are said to be denegrate?
Thanks!
(No, this isn’t homework in case you are wondering. Tis curosity!)
There’s nothing special about finding degenerate critical points. (BTW, it’s “degenerate”, not “denegrate”.) You just find all the critical points, and the ones where the Hessian is zero are the degenerate ones.
The trick is classifying the degenerate critical points, and unfortunately there’s no good formulaic way to do that. First of all, there are many, many different possibilities: a degenerate critical point can be a local max, a local min, a saddle point, a “monkey saddle”, something like the side of a cylinder, or probably other things I haven’t even thought of. If you need to get some idea of what a degenerate critical point is, you can try to take the Taylor expansion of the function at that point and see what the first non-zero terms are, but even for two-variable functions that gets really messy really fast. Other than that, the best you can do is probably try and plot the function via computer and see if that gives you a clue.
You’re right. I actually did mean “classifying” critical points. I really wasn’t thinking when I wrote this post.
Anyhow, you mentioned several things the degenerate points could be. Local Maximums, Minimums, and Saddle Points I have understood. But the others that you have listed have not yet come up in my class. Are we simply not using enough variables? Not the correect formatting? (FYI, so far I’ve only been using x, y, and z. No u yet).
An example of a “monkey saddle” is the graph of f(x,y)=x[sup]3[/sup]-3xy[sup]2[/sup], near the point (x,y)=(0,0). If you plot that graph on your friendly neighbourhood graphical calculator or computer, and compare it to a typical saddle point, you’ll get some idea of where the name came from. (Silly name really, but it stuck.)
And by “something like the side of a cylinder”, all I meant was something like the graph of f(x,y)=sqrt(1-x[sup]2[/sup]) near (x,y)=(0,0), for example. That’s sort of a local maximum, but not a strict local maximum, since f(0,y)=f(0,0) for all y.
So no, you don’t need any more variables or anything. The point is that this is just the tip of the iceberg. When the Hessian equals zero, all bets are off.