I've recently started doing some multivariable calculus questions, and I've come across questions such as this:

  • If $f: \mathbb{R}^2 \to \mathbb{R}$ has only one stationary point and it is a local max, is it a global max?

  • Find a function all of whose directional derivatives are $0$ at $0$ but is unbounded in any neighborhood of $0$.

I have great difficulty with these problems, because I have very little intuition about graphs in 3D. Here is an answer to the first question. This is exactly the type of intuition I would like to develop:

Loose description of the geometry: imagine a flat plane and then you put a lone hill with a peak on it. Now tilt the plane a little. Now you still have one peak, but hopefully you can also see that you have introduced a saddle point. Imagine sliding the saddle point location off to infinity to make it effectively no longer there.

Is there any textbook which focuses on this 3D intuition in analysis?

  • $\begingroup$ I'm not aware of a book that directly helps with this. Enough years of experience teaching out of different books has exposed me to exercises that happen to have the features of the two questions you posed. But the textbook author is providing the example function and the exercise is to show that it has those interesting features. That's different than posing the question about if these things exist, and having a strategy to find them. $\endgroup$
    – 2'5 9'2
    Jan 16, 2021 at 23:35
  • $\begingroup$ What I said with that "loose description" is not me using my imagination to visualize the thing ahead of time. Rather, I could recall what it looked like (after an author once upon a time had exposed me to an example) and that is my attempt to describe it. $\endgroup$
    – 2'5 9'2
    Jan 16, 2021 at 23:37
  • $\begingroup$ With your second question, experience also tells me how to construct such an example. It's easiest to describe a discontinuous example, which appears legal based on your question. Make the function 0 everywhere. But take a path that curves its way through the origin. For example y=x^2. Along that path, redefine the function to spike to infinity. For example at (t,t^2), define the output to be 1/t. (Leave the value to be 0 for t=0.) Directional derivatives at the origin are unaffected, because in any direction there is still at least a tiny interval along which the function is constantly 0. $\endgroup$
    – 2'5 9'2
    Jan 16, 2021 at 23:46
  • $\begingroup$ @alex.jordan Thank you so much for the insightful and honest comments! $\endgroup$
    – user56202
    Jan 17, 2021 at 0:11
  • $\begingroup$ Hmmm ... Looks familiar. $\endgroup$ Jan 17, 2021 at 1:04


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