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Math Help - method of Lagrange multipliers question

  1. #1
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    Question method of Lagrange multipliers question

    Can someone tell me how to use method of Lagrange multipliers to do following question:

    Use the method of Lagrange multipliers to find the critical points of the following constrained optimisation problems. Do not attempt to test the nature of these critical points.

    (a) Optimise f(x, y) = x + y,
    subject to the constraint: x^2 + y^2 - 4 = 0

    (b) Optimise f(x, y, z) = x^2 + y^2 + z^2,
    subject to the constraints: x + y + z = 1 and x + 2y + 3z = 2
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  2. #2
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    Quote Originally Posted by quah13579 View Post
    Can someone tell me how to use method of Lagrange multipliers to do following question:

    Use the method of Lagrange multipliers to find the critical points of the following constrained optimisation problems. Do not attempt to test the nature of these critical points.

    (a) Optimise f(x, y) = x + y,
    subject to the constraint: x^2 + y^2 - 4 = 0

    (b) Optimise f(x, y, z) = x^2 + y^2 + z^2,
    subject to the constraints: x + y + z = 1 and x + 2y + 3z = 2

    (a)

     f = x+y + a ( x^2 + y^2 - 4 )

    we have

     f_x = 1 + 2ax = 0 and

     f_y = 1 + 2ay = 0

     x = y = - \frac{1}{2a}

    Sub.  x=y into the constraint  x^2 + y^2 = 4

     2x^2 = 4 , x = \sqrt{2}  ~ or~ -\sqrt{2}

     (x,y) = (  \sqrt{2}, \sqrt{2}) ,( -\sqrt{2}, \sqrt{2}) ,(  \sqrt{2}, -\sqrt{2}),(  -\sqrt{2}, -\sqrt{2})

    (b)

     f= x^2 + y^2 + z^2 + a( x + y + z -1 ) + b( x + 2y + 3z -2)


     f_x = 2x + a + b = 0 (1)
     f_y = 2y + a + 2b = 0 (2)
     f_x = 2z + a + 3b = 0 (3)

     x = -\frac{1}{2} (a+b)
     y = -\frac{1}{2} (a+2b)
     z = -\frac{1}{2} (a+3b)

    It contructs a plane , also from the two constraints , we have three planes now , solve the system of three linear equations .


     (x,y,z) = ( \frac{1}{3} , \frac{1}{3} , \frac{1}{3} )
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  3. #3
    Super Member 11rdc11's Avatar
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    Quote Originally Posted by quah13579 View Post
    Can someone tell me how to use method of Lagrange multipliers to do following question:

    Use the method of Lagrange multipliers to find the critical points of the following constrained optimisation problems. Do not attempt to test the nature of these critical points.

    (a) Optimise f(x, y) = x + y,
    subject to the constraint: x^2 + y^2 - 4 = 0

    (b) Optimise f(x, y, z) = x^2 + y^2 + z^2,
    subject to the constraints: x + y + z = 1 and x + 2y + 3z = 2
    A)

    1 = \lambda 2x

    1 = \lambda 2y

    x^2 + y^2 = 4

    x = \frac{1}{2\lambda} = y

    so plug into your other equation

    x^2 + x^2 =4

    2x^2 =4

    x^2 = 2

    x = ^+_- \sqrt{2}

    so your points are

    (\sqrt{2},\sqrt{2}),(-\sqrt{2},\sqrt{2}),(-\sqrt{2},-\sqrt{2}),(\sqrt{2},-\sqrt{2})

    Now just see which is you maxs and mins.

    Try following my approach for the next problem.
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