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Math Help - Lagrange Multiplier

  1. #1
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    Lagrange Multiplier

    Use the method of Lagrange multipliers to find the min value of:

    f(x, y) = 5x^2 - 4x^2 subject to the constraint g(x, y) = x^2 + y^2 = 9

    Here's what I have so far:

    f_x = 10x
    f_y = -8y
    g_x = 2x
    g_y = 2y

    10x = 2 \lambda x
     -8y = 2 \lambda y
    9 = x^2 + y^2


    \frac{5x}{x} = \lambda

    -\frac{4y}{y} = \lambda

    \frac{5x}{x} = -\frac{4y}{y}

    5xy = -4xy

    and now... I don't know how to isolate for x or y.
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  2. #2
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    Quote Originally Posted by Macleef View Post
    Use the method of Lagrange multipliers to find the min value of:

    f(x, y) = 5x^2 - 4x^2 subject to the constraint g(x, y) = x^2 + y^2 = 9

    Here's what I have so far:

    f_x = 10x
    f_y = -8y
    g_x = 2x
    g_y = 2y

    10x = 2 \lambda x (**)
     -8y = 2 \lambda y
    9 = x^2 + y^2


    \frac{5x}{x} = \lambda

    -\frac{4y}{y} = \lambda

    \frac{5x}{x} = -\frac{4y}{y}

    5xy = -4xy

    and now... I don't know how to isolate for x or y.
    I'll pick it up at this point (**). From these you have

    2x(\lambda - 5) = 0,\;  2y(\lambda+4) = 0 subject to 9 = x^2 + y^2.

    From the first you have x = 0\; \text{or}\, \lambda= 5

    and from the second

    y = 0\; \text{or}\, \lambda= -4 noting that x = 0, y = 0 cannot both happen due to the constraint.

    If x = 0\;\; \text{then}\; y= \pm 3 (from the constaint)

    or y = 0\;\; \text{then}\; x= \pm 3 (from the constaint)

    Now choose the one that give the min.
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  3. #3
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    Quote Originally Posted by Macleef View Post
    Use the method of Lagrange multipliers to find the min value of:

    f(x, y) = 5x^2 - 4x^2 subject to the constraint g(x, y) = x^2 + y^2 = 9

    Here's what I have so far:

    f_x = 10x
    f_y = -8y
    g_x = 2x
    g_y = 2y

    10x = 2 \lambda x
     -8y = 2 \lambda y
    9 = x^2 + y^2


    \frac{5x}{x} = \lambda (**)

    -\frac{4y}{y} = \lambda

    \frac{5x}{x} = -\frac{4y}{y}

    5xy = -4xy

    and now... I don't know how to isolate for x or y.
    On a side note, the step (**) is ok provided that x \ne 0 so this would have to be considered as a special case.
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