# Maximize multivariable function with infinite maxima

1. Aug 14, 2014

### 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

2. Aug 14, 2014

### 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?

3. Aug 14, 2014

### 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.

4. Aug 15, 2014

### 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.