Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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?


    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)!

  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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook