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Math Help - Local Maxima or Minima

  1. #1
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    Local Maxima or Minima

    I need to determine whether the fct has any local minimums and/or maximums.

    f(x,y)=x^2-2xy+y^2

    I optimized and found that the f.o.c gives: x=y (right?). Does that mean that at ALL constants, c, where x=y=c, we have a local minima? Or does this mean that the function doesn't have any optimal values?

    tks
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  2. #2
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    First of all, note that x^2 - 2 x y + y^2 = (x-y)^2. As you should see, the function cannot obtain negative values. Setting x=y gives a function value of 0, which obviously is a minimum for all x=y. You might also want to have a look at the second plot here: http://www.wolframalpha.com/input/?i=x^2-2+x+y+%2B+y^2
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  3. #3
    Super Member Failure's Avatar
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    Quote Originally Posted by altave86 View Post
    I need to determine whether the fct has any local minimums and/or maximums.

    f(x,y)=x^2-2xy+y^2

    I optimized and found that the f.o.c gives: x=y (right?). Does that mean that at ALL constants, c, where x=y=c, we have a local minima? Or does this mean that the function doesn't have any optimal values?

    tks
    Practically speaking, this problem ist a joke: since f(x,y)=x^2-2xy+y^2=(x-y)^2\geq 0 you can immediately say that f assumes its global minimum 0 at all points (c,c),\;c\in\mathbb{R}.

    Speaking more theoretically: with x=y you have only found a necessary condition for f assuming a locally extreme value at all points (c,c),\;c\in\mathbb{R}. Now, in principle, you would have to examine the behavior of the second derivative.
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  4. #4
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    Quote Originally Posted by Failure View Post
    this problem ist a joke
    Got you! Another German speaker here. Can't remember how often that happened to me.
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