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Fixed points and graphs

  1. Mar 27, 2006 #1
    Hi, I was thinking about the following and would like some clarification. Suppose that we have a continuous scalar function [itex]f:R^n \to R[/itex] with a critical point at say x_0, where the dimension of x_0 depends on the value of n.

    Consider as an example f(x) = x (n = 1). The point x = 0 is a critical point since f'(x) is zero at that point. Since f is continuous then corresponding to x = 0 must be a local minimum, local maximum or saddle correct? (Not exactly sure about it)

    My point is that x = 0 lies on a line of fixed points and hence cannot correspond to a maximum or a minimum? Is this true in higher dimensions or does this reasoning hold at all?

    Any help would be good thanks.
  2. jcsd
  3. Mar 27, 2006 #2
    if f(x) = x, then f'(x) = 1 for all x. so x=0 is not a critical point. f is an increasing function.
    Last edited: Mar 27, 2006
  4. Mar 27, 2006 #3


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    As nocturnal said, if f(x)= x then x= 0 is NOT a critical point. Maybe you were thinking of f'(x)= x which would correspond to f(x)= (1/2)x2+ C. That really does have a minimum at x= 0.
  5. Mar 29, 2006 #4
    Oops, I should watch my differentiation...better hope that doesn't happen during an exam.:biggrin:
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