Consider the picture below:

Level curves for a function f(x,y)

This is the levels curves for a function $f(x,y)$ where:

  • Blue line is the partial derivative of $f(x,y)$ with respect to x
  • Red line is the partial derivative of $f(x,y)$ with respect to y

I know that the intersection of the partials are crtical points. However I do not know how to identify other points which might be crtical but not be local max/minimi points. Like the saddle point, what should I look for?

  • $\begingroup$ Where did you learn that the intersection of partials is a critical point? There is a critical point if and only if the partials are equal to zero (the gradient). $\endgroup$ – Bor Kari Apr 14 at 1:16
  • $\begingroup$ @BorKari frogot to add that both are equal to zero. So where they intersect both are zero at that point, meaning that we have a critical point. That was my reasoning, but now i'm not as sure $\endgroup$ – oxodo Apr 14 at 1:27

You should check the second order derivative.

Second derivative test

  • $\begingroup$ I am unfortunately supposed to use this level curve to find points of interest. I am able to indentify points, local max/min, using this picture for level curves. But I don't know what to look for when it comes to saddle points. I also want to add that the levels curves for the partial derivatives are when they are equal to zero. $\endgroup$ – oxodo Apr 14 at 15:33

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