Find maximum of a double integral over a region I have a region given by $$R = |{ax}|+|{by}| \le 1$$ and $$f(x,y) = \iint\limits_{R}{(ax-by)^2 \ \cdot \ (3ab^3+12a^3b-6a^3b^2) \ \cdot \ \sin^2({\pi ax + \pi by}})dxdy$$
I need to find the values of $a$ and $b$ that maximize $f$ and I have no idea where to start. 
 A: Assuming that this integrand is typed correctly (I am wondering about this for reasons that will be made clear ultimately), the double integral is actually something of a "McGuffin" -- it has a small part in the maximization, but its exact value is not important.
Since $ \ a \ $ and $ \ b \ $ are (presumably positive) constants, we will re-write the integral as  
$$  (3ab^3+12a^3b-6a^3b^2) \iint\limits_{R}{(ax-by)^2 \ \cdot \    \sin^2({\pi [ ax +   by ] }}) \ \ dx \ dy \ \ . $$
The region of integration is a rhomboid (or lozenge)  and, as Poppy suggests, a substitution would be in order.  It will be perhaps cleanest to use  $ \ u \ = \ ax \ + \ by \ $ and $ \ v \ = \ ax \ - \ by \ $ , which has the Jacobian  
$$ \mathfrak{J} \ = \ \left| \begin{array}{cc }
\frac{1}{2a} & \frac{1}{2a} \\ \frac{1}{2b} & -\frac{1}{2b} \end{array} \right| \ \ .  $$

The integral becomes
$$  3ab \ (b^2+4a^2-2a^2b ) \int_{-1}^1 \int_{-1}^1 \ {v^2 \ \cdot \  | - \frac{1}{2ab} | \ \cdot \   \sin^2({\pi u }}) \ \ dv \ du \ \ . $$
As the Jacobian is negative, the orientation of the transformed square is reversed relative to the original rhombus.  The double integral proves to have a value of  $ \ \frac{2}{3} \ $ , but this is of no import.  The significant factor comes from the Jacobian determinant, so that we now wish to maximize
$$  \frac{3}{2} \ (b^2+4a^2-2a^2b )   \cdot \   \frac{2}{3} \ \ . $$
Now here is where the question arises.  This function is minimized for $ \ a \ = \ 0 \ \ , \ \ b \ = \ 0 \ $ (giving zero for the function), but the other critical points are the saddle points $ \ a \ = \ \pm \sqrt{2} \ \ , \ \ b \ = \ 2 \ $ , where the function has the value 4  .  So I am not sure the expression given is what was intended; simple alterations don't fix the "saddle point" difficulty.
