Purchase Solution

Real Analysis : Finding a Maximum using Lagrange Multipliers

Not what you're looking for?

Ask Custom Question

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.

Attachments
Purchase this Solution
Solution Summary

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

Solution Preview

Please see the attached file for the complete solution.
Thanks for using BrainMass.

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)
g(x1,x2,x3,x4)=1
h(x1,x2,x3,x4) =0

This is a system of 6 equations with 6 unknowns
grad(F)=(1,1,1,1)
grad(g)=(2x1,2x2,2x3,2x4)
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 ...

Purchase this Solution
Free BrainMass Quizzes
Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Probability Quiz

Some questions on probability

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

View More Free Quizzes
Related BrainMass Solutions

  • Finding Extrema: Lagrange Multipliers

    23173 Finding Extrema: Lagrange Multipliers Use the method of Lagrange multipliers to find the indicated extremum. Let f(x,y) = 8x^2 - 24xy + y^2.

  • Lagrange multipliers method

    438979 Lagrange multipliers method Considering the surface f(x,y)=xy and the constraint x2+y2=1 , answer to the following questions: A. Using the Lagrange multipliers method we can obtain some possible maximum and minimum for ?=ï?±1/2 B.

  • Lagrange Multipliers to find maximum

    109515 Lagrange Multipliers to find maximum Using Lagrange multipliers, find the maximum and minimum values of f(x,y) = x^2*y subject to the x^2 + y^2 = 1 constraint. Please see the attached file for the complete solution.

  • Lagrange multipliers

    207117 Using Lagrange Multipliers to Find Extreme Values Use Lagrange multipliers to find maximum and minimum values of the function subject to a given constraint or constraints. 1.)f(x,y,z)= xyz; x^2 + 2y^2 + 3z^2 = 6 2.)f(x,y,z)= x^2 + y^2 + z

  • The Method of Lagrange Multipliers

    23174 The Method of Lagrange Multipliers Use the method of Lagrange multipliers to find the indicated extremum. You may assume the extremum exists.

  • Minimization and contour point

    Compare your answers with those in the part (a). Solution. Using Lagrange multipliers, we first need to form a function as follows. Now we find the partial derivatives.

  • Constrained Minimization using Lagrange Multipliers

    111976 Constrained Minimization using Lagrange Multipliers Please see attached PDF file; I'm trying to derive a couple of results involving Lagrange multipliers and a linear algebra equivalence.

  • Global maximum of given set and extreme values

    The solution gives detailed steps on finding the global maximum on the given set and extreme values using Lagrange multipliers. Step by step calculations are provided.

  • Partial derivative and double integral

    22467 Partial Derivative and Double Integral The problems are attached 1 -5 based on Chapter Partial Derivative - (Maximum & Minimum Values and Lagrange Multipliers 1.

View More Related Solutions