Max/Min of a function using Lagrange Multipliers

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To find the absolute maximum and minimum values of the function f(x,y) = sin x + cos y on the rectangle defined by 0<=x<=2pi and 0<=y<=2pi using Lagrange Multipliers, the constraints must be properly incorporated. The method involves setting up the equations based on the gradients of the function and the constraints. A common approach is to express the constraints in terms of a constant, such as x+y=C, but this does not directly apply to the rectangular boundaries. It is essential to evaluate the function at the critical points found within the interior and along the boundaries of the rectangle. The solution requires careful consideration of both the function's behavior and the constraints to identify the extrema accurately.
andrewjohnsc
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Homework Statement


Find the absolute maximum and minimum values for f(x,y) = sin x + cos y on the rectangle R defined by 0<=x<=2pi and 0<=y<=2pi using the method of Lagrange Multipliers.



The Attempt at a Solution


I don't know where to start in getting the constraint into something I can work with! :confused:
 
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you can always use x+y=C where C is some constant.
 
I don't understand how that constrains the function to 0<=x<=2pi 0<=y<=2pi
 

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