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2 var critical point questions

  1. Dec 18, 2004 #1

    quasar987

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    I have few questions about extrema of fonctions of two variables. It is written in my textbook: "At a local maximum, the gradient vector must be nul or undefined. A similar reasoning shows that the gradient must be nul at a local minimum." Actually there was no preceeding reasoning to this statement so I don't understand.

    - Why couldn't the gradient vector be undefined at the min?

    - If one of the partial derivative is undefined at a certain point, does it automatically means the point is a max? If no, how do you tell analytically?

    - If (a,b) is a critical point because the gradient at (a,b) is 0 and if the test of the second order partial derivative fails (i.e. =0). How can I conclude analytically to the nature of the critial point?
     
    Last edited: Dec 18, 2004
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  3. Dec 18, 2004 #2

    arildno

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    Dearly Missed

    The author has been sloppy!
    - Why couldn't the gradient vector be undefined at the min?
    It certainly can be!
    .
     
  4. Dec 18, 2004 #3

    quasar987

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    Ok, then I will rewrite question #2:

    - If one of the partial derivatives is undefined at a certain point, how do I conclude analytically to the nature of the critical point?

    - Could it be that a derivative in a direction other than the x or y axis is undefined while it is defined in the direction of the x and y axis? In this case wouldn't the method of analysis of the partial derivatives fail to detect the critical point?
     
  5. Dec 18, 2004 #4

    Hurkyl

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    Remember that the definition of optima don't involve derivatives at all -- when differential techniques fail, you often have to resort to the definitions to get your answers.
     
  6. Dec 18, 2004 #5

    arildno

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    And, I would think, some functions might be perverse enough to refuse yielding up where its extrema are, despite our best efforts..
     
  7. Dec 18, 2004 #6

    Hurkyl

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    Even worse, there are functions that don't even have local extrema!

    example: (here, p and q are relatively prime)


    f(x) = 0 if x is irrational
    f(p/q) = 1 - 1/q if q is even
    f(p/q) = -1 + 1/q if q is odd
     
    Last edited: Dec 18, 2004
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