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Trouble proving a function with 2 variables does not have a global max or global min

  1. Jul 26, 2012 #1
    I am struggling with this question which appears in every adv. calculus exam paper I practise and would love some help or advice on how to even approach it! I have no trouble getting the extreme points and determining whether they are local minimum, local maximum or saddle points, but proving that a function with 2 variables does not achieve a global max or min is proving very difficult. Here is an example of a question:

    1. Let f(x,y)=y^2+2xy+x^3-x. Find the critical points of f and classify each of them as a local maximum, a local minimum or a saddle point.

    (The answers I have come up with for this part are : (-1/3, 1/3) is a saddle point and (1,-1) is a local minimum.

    2. Consider the values of f on the x- axis, or otherwise, to show that f has neither a global maximum nor a global minimum.

    I don't know what to do here, especially since I am not given an interval...
     
  2. jcsd
  3. Jul 26, 2012 #2

    tiny-tim

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    welcome to pf!

    hi porroadventum! welcome to pf! :smile:
    the general strategy is to consider the turning points and the whole of the boundary

    if it has a maximum or minimum, it must either be at a local maximum or minimum, or it must be on the boundary
    (try using the X2 button just above the Reply box :wink:)

    you are given an interval …

    it's the whole plane!! :biggrin:

    you have to consider the boundary, which is "at infinity" in every direction

    in this case, if you follow the hint and put y = 0, you should easily prove that it reaches both +∞ and -∞ "at infinity" :wink:
     
  4. Jul 26, 2012 #3
    Re: Trouble proving a function with 2 variables does not have a global max or global

    Sorry I can be a bit dim sometimes, so to make sure I've understood:-

    BY letting y=0 I get the function x^3-x and then I calculate the limit of the function as x approaches +∞ (=+∞) and -∞ (=-∞). If the limits were to equal a real number, would this mean that this real number is the global max/ min?

    (I'm really sorry if I've gotten it completely wrong...)
     
  5. Jul 26, 2012 #4
    Re: Trouble proving a function with 2 variables does not have a global max or global

    *x3-x not x^3-x, woops!
     
  6. Jul 26, 2012 #5

    tiny-tim

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    it could be, but you would have to compare that value with the values at the local maxima and minima … to see who wins! :biggrin:

    ohh, and of course with the values at every other direction "at infinity" (not just the x and y axes :wink:)
     
  7. Jul 26, 2012 #6
    Re: Trouble proving a function with 2 variables does not have a global max or global

    OK, thank you so much for your help! I am so grateful!
     
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