Maximize multivariable function with infinite maxima

[Patrick94]

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

 

[da_nang]

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?

 

[Patrick94]

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.

 

[da_nang]

In general, you can try the second partial derivative test.

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

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