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Real Analysis : Finding a Maximum using Lagrange Multipliers

Please see the attached file for the fully formatted problem.

What is the maximum of F = x1 +x2 +x3 +x4 on the intersection of x21 +x22 +x23 + x24 = 1 and x31+ x32+ x33+ x34= 0?
As this is an analysis question, please be sure to be rigorous and as detailed as possible.


Solution Summary

LaGrange Multipliers are employed to turn constraints into six equations and six unknowns. The solution is detailed and well presented.