Optimization/maximization with multivariable calculus

1. Nov 26, 2007

1. The problem statement, all variables and given/known data

Maximize the Riemann Sum $$\Sigma$$ $$^{n}_{i=1}$$ x$${i}$$*y$${i}$$ subject to constraints $$\Sigma$$ $$^{n}_{i=1}$$ x$$^{2}_{i}$$=1 and $$\Sigma$$ $$^{n}_{i=1}$$ y$$^{2}_{i}$$=1
2. Relevant equations
My teacher doesn't speak English very well. I'm in Calculus 3 and the average on his exams are around 40%. I'm a good Engineering student and need help. In class, we did an optimization problem where you can maximize the volume of say a box since you know length, width, height. We never went over this and I'm wondering how to do this. Despite him teaching us derivatives in that section, should I approach by doing integration and find the bounded region of x^2 and y^2 individually? Limits were never taught that much in high school so I really would appreciate your help.

3. The attempt at a solution
Maybe try the partial derivatives of the last two equations with respect to x and y, then set them to zero to find critical points? That's the only way I knew how to find local max and absolute max. Please help if you can. Thanks guys.