Optimization/maximization with multivariable calculus

[adradmin]

Homework Statement

Maximize the Riemann Sum $\Sigma^{n}_{i=1}x_{i}*y_i$ subject to constraints $\Sigma^{n}_{i=1}x^{2}_{i}=1$ and $\Sigma^{n}_{i=1}y^{2}_{i}=1$

Homework 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.

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.

 

[haruspex]

You could try Lagrange multipliers, but in this case the easiest is probably to think what it means in terms of vectors. What should the relationship be between $\vec x$ and $\vec y$?

 

[Mark44]

I assume it should read <snip>

So do I, and I have edited the OP to read the same as what you wrote. I doubt we'll hear from the thread starter, who hasn't been back since 2007.

 

[haruspex]

I doubt we'll hear from the thread starter, who hasn't been back since 2007.

Yes, it is from the tranche of old unanswered threads that came in from MHB. My approach to these is to answer exactly as I would for a fresh thread.