# Bivariate function maximization

I have a bivariate function like

$f(x,y) = \frac{1}{x^3 \sqrt{\pi}}. e^{\frac{2-x}{x^2}} . y^3 . e^{3.y \over 3-y}$

and I want to find its global maximum over a range of $x \in [0, 200] \text{, and } y \in [300,50000]$

What kind of algorithms I can use to find the global maximum. I want to have keywords for searching and finding materials to study

Is there any java library that I can use to solve these kind of problems?

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Find the critical points first, i.e., $\displaystyle\{(a,b):\left.\frac{\partial f}{\partial x}\right|_{(a,b)}=\left.\frac{\partial f}{\partial y}\right|_{(a,b)}=0\}$ Among those points find the points where second order derivatives $\frac{\partial^2f}{\partial x^2}$ and $\frac{\partial^2f}{\partial y^2}$ are negative. Also need to compute the values on the boundary of the region, since we need global maximum. I can't proceed from here! Does the mixed partial derivative of order 2 come in the picture? – Host-website-on-iPage Jul 13 '12 at 17:32
The example you give is of the form $f(x,y)=g(x)h(y)$. This makes things simpler. For a genuinely bivariate function you have to determine the stationary points in the interior of the given rectangle and the conditionally stationary points on the four sides. In this way you obtain a list of candidates, whereupon the global extrema can be determined by comparison of values. – Christian Blatter Jul 13 '12 at 17:40
@Aneesh Yes they do. See en.wikipedia.org/wiki/Second_partial_derivative_test – user31373 Jul 15 '12 at 4:53