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Real Analysis : Finding a Maximum using Lagrange Multipliers

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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.

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Solution Summary

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

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Find maximum of F(x)=x1+x2+x3+x4 on the intersection of
x1 2+x2 2+x3 2+x4 2=1 and
x1 3+x2 3+x3 3+x4 3=0

I used Lagrange Multipliers Method. We have 2 constraints: x1 2+x2 2+x3 2+x4 2=1 and
x1 3+x2 3+x3 3+x4 3=0 . Call them g (x1,x2,x3,x4) and h(x1,x2,x3,x4)
We need to solve the following system of equations:

grad(F)=a grad(g)+b grad(h)
h(x1,x2,x3,x4) =0

This is a system of 6 equations with 6 unknowns
grad(h)=(3x1 2,3x2 2,3x3 2,3x4 2)

And so we have
1=a2x1+b3x1 2
1=a2x2+b3x2 2
1=a2x3+b3x3 2
1=a2x4+b3x4 2
x1 2+x2 2+x3 2+x4 2=1
x1 3+x2 3+x3 3+x4 3=0

Subtracting 2nd from 1st equations, 3rd from 2nd, 4th from 3rd and 1st from 4th we get
equivalent system:
a2(x1-x2)+b3(x1 2-x2 2)=0
a2(x2-x3)+b3(x2 2-x3 2)=0
a2(x3-x4)+b3(x3 2-x4 2)=0
a2(x4-x1)+b3(x4 2-x1 2)=0
x1 ...

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