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Maximize multivariable function with infinite maxima

  1. Aug 14, 2014 #1
    Could someone walk me through how to maximize this 2-variable function wrt z?

    http://www.wolframalpha.com/input/?...)^2)))+-+100/(1+(root+((x-2)^2+++(y-3)^2))^2)

    I know the set of solutions will form a circle around the point (2,3). How do I go about finding the set of maxima that form this circle/the equation of this circle?

    (I am a complete math novice)!

    Thanks
     
  2. jcsd
  3. Aug 14, 2014 #2
    Well, if you know that the set of solutions form a circle, then you can transform the problem to one dimension by changing to a polar coordinate system, no?
     
  4. Aug 14, 2014 #3
    I want to be able to solve in Cartesian coordinates, I think, since this is the very simplified form of a function which will contain many more terms.
     
  5. Aug 15, 2014 #4
    In general, you can try the second partial derivative test.

    Let [itex]\vec H_k(f(\vec x))[/itex] be the Hessian matrix of the function [itex]f(\vec x)[/itex] (evaluated at [itex]\vec x[/itex]) of the [itex]k[/itex] first variables, where [itex]k = 1, 2, 3, ... , n[/itex].

    If you're function is [itex]f(\vec x)[/itex] then the critical point [itex]\vec p[/itex], i.e. [itex]\nabla f(\vec p) = \vec 0[/itex], is a local minimum if [itex]\forall k : |\vec H_k(f(\vec p))| > 0[/itex] and a local maximum if [itex]\forall k : (-1)^k |\vec H_k(f(\vec p))| > 0[/itex]. For all other cases, [itex]\vec p[/itex] is a saddle point unless [itex]|\vec H_n(f(\vec p))| = 0[/itex], for which the test is inconclusive.
     
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