Saddles, who needs them?

It is not difficult to prove that every continuously differentiable function f:\mathbb{R}\to\mathbb{R}:

  • Must have a minumum point between two maximums and –
  • If it has precisely one critical point, be it a local minimum or a local maximum, that point must also be a global minimum or maximum.

What is amazing that neither of these statements holds in two dimensions. There are relatively simple examples of infinitely smooth functions of two variables that have just two critical points, both maximums, with no saddle or minimum between them, or have just one critical point, which is a local minimum, but not the global one.

These examples are pretty recent – that is to say they were discovered only about 20 years ago. They took my breath away when I learned about them – I couldn’t believe something like that could possibly exist, and yet it did. Here they are:

The first example is also the neatest one: f(x, y)=x^2(1-y)^3+y^2. It gets bonus weirdness points for being a polynomial, of all things. Here’s what it looks like:

You can clearly see the one and only critical point: a local minimum which at the same time is clearly not global.

The second example might be a bit easier to understand. Here we have a local-but-not-global maximum: g(x, y)=3xe^y-x^3-e^{3y}

Perhaps the most disturbing of the three will be the last function, with two maximums and no saddle between them: h(x, y)=-(x^2y-x-1)^2-(x^2-1)^2  

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