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Saddle points of functions of n variables

  1. Apr 22, 2014 #1

    I need some clarification on exactly which critical points are saddle points. The definition I am finding everywhere is that "A saddle point is a stationary point that is neither a local minimum nor a local maximum." My question is: what kind of minimum and maximum is the above definition about? The strict or the non-strict kind?

    In case that strict extrema are implied, then, for example, the constant function f(x, y) = c has a saddle point at each point of its domain, whereas in the case of non-strict extrema, the same function has no saddle points, because at every point it has both a local non-strict minimum and a local non-strict maximum. A similar ambiguity arises in the case of, for example, the function f(x, y) = x^2. Are all points that lie on the x = 0 line saddle points since they are non-strict extrema?

    Thanks for reading :)
    Last edited: Apr 22, 2014
  2. jcsd
  3. Apr 22, 2014 #2
    As I understand it, a saddle point is a critical (stationary) point that is not a local (non-strict) extreme point.
    So you are correct. In your first example, all points in your domain are saddle points. In your second example all points along the line x=0 are saddle points.
  4. Apr 22, 2014 #3
    Thanks :) After a bit more googling I found these lecture notes where first a distinction is drawn between strict and non-strict extrema and then a saddle point is defined as you too understand it. So I guess my question is answered!
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