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Strict local minimizer (multivariate)

  1. Oct 19, 2004 #1
    I'm stuck on this question

    Show that f(x1,x2) has a strict local minimizer at t=0 along every line

    { x1=at
    { x2=bt


    through (0,0).


    Any hints or tips would be great thanks
     
  2. jcsd
  3. Oct 20, 2004 #2

    HallsofIvy

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    What exactly do you mean by "minimizer"? I'm not familiar with that term.

    At first I thought you meant "minimum" but in that case the theorem is not true.

    Suppose f(x,y)= (x-1)2+ (y-1)2.

    Saying that f has a strict local minimum at t= 0 on x= at, y= bt would simply mean that f has a strict local minimum at (0,0) but that is not true- the only minimum of f is at (1,1).
     
  4. Oct 20, 2004 #3
    The minimizer is the point t where the minimum is. THat's why I'm a bit confused with the question. The wording I was given in my book is a bit awkward.

    I think what it means is. For every f(x1,x2) given that x1 and x2 are lines.

    There is a minimum at t=0

    Those that make sense?
     
  5. Oct 21, 2004 #4

    HallsofIvy

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    No! In particular, the sentence "For every f(x1,x2) given that x1 and x2 are lines." makes no sense at all. x1, x2 are variables that I presume are numbers, not lines.

    Aren't there some conditions on the function f(x1, x2)? The statement is certainly NOT true for general f.
     
  6. Oct 21, 2004 #5
    I think you're right. The wording of the question is not very good. A strict local minimizer is in fact the same as a strict local minimum.

    I geuss I'll have to ask the prof
     
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