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Math Help - Lagrange Multipliers

  1. #1
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    Lagrange Multipliers

    F(x,y,z)=4x^2+y^2+5z^2. Use Lagrange Multipliers to find the point on the plane 2x+3y+4z=12 at which F(x,y,z) is the least value.

    Δf(x,y,z)= 8xi+2yj+10zk
    γΔg(x,y,z)= 2λ+3λ+4λ

    8x=2λ
    3y=3λ
    10z=4λ

    Im stuck
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  2. #2
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    Quote Originally Posted by JJ007 View Post
    F(x,y,z)=4x^2+y^2+5z^2. Use Lagrange Multipliers to find the point on the plane 2x+3y+4z=12 at which F(x,y,z) is the least value.

    Δf(x,y,z)= 8xi+2yj+10zk
    γΔg(x,y,z)= 2λ+3λ+4λ

    8x=2λ
    3y=3λ
    10z=4λ

    Im stuck
    use your fourth equation and try to work out by multiplying each equation by the missing variables.
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  3. #3
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    Quote Originally Posted by Warrenx View Post
    use your fourth equation and try to work out by multiplying each equation by the missing variables.
    Fourth equation, you mean γΔg(x,y,z)= 2λ+3λ+4λ? How do I solve for λ?
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  4. #4
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    Quote Originally Posted by JJ007 View Post
    F(x,y,z)=4x^2+y^2+5z^2. Use Lagrange Multipliers to find the point on the plane 2x+3y+4z=12 at which F(x,y,z) is the least value.

    Δf(x,y,z)= 8xi+2yj+10zk
    γΔg(x,y,z)= 2λ+3λ+4λ

    8x=2λ
    3y=3λ
    10z=4λ

    Im stuck
    might be wrong, but here goes:
    g= lambda
    since 8x = 2g
    that means that 8 = 2g/x
    which means that 12 = 6g/2x
    so 6g/2x = 2x +3y + 4z
    6g = 2x(2x + 3y + 4z)
    6g = 1/2g(1/2g + 3g + 4z)
    12g^2 -3.5g = 4z
    10z = 4g
    z = 2/5g
    8/5g + 3.5g =12g^2
    1.6g + 3.5g = 5.1g
    5.1g/g
    5.1 = 12g

    go from there.. i think
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  5. #5
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    Hello, JJ007!

    f(x,y,z)\:=\:4x^2+y^2+5z^2

    Use Lagrange Multipliers to find the point on the plane 2x+3y+4z\:=\:12
    at which f(x,y,z) is a minimum.

    We have: . F(x,y,z,\lambda) \;=\;(4x^2 + y^2 + 5z^2) + \lambda(2x + 3y + 4z - 12)



    Set the partial derivatives equal to zero.

    . . \begin{array}{ccccccc}F_x \:=\:8x + 2\lambda \:=\:0 & \Rightarrow & x \:=\:-\dfrac{1}{4}\lambda & [1] \\ \\[-3mm]<br /> <br />
F_y \:=\:2y + 3\lambda \:=\:0 & \Rightarrow & y \:=\:-\dfrac{3}{2}\lambda & [2]\\ \\[-3mm]<br /> <br />
F_z \:=\:10z + 4\lambda \:=\:0 & \Rightarrow & z \:=\:-\dfrac{2}{5}\lambda & [3]\end{array}

    . . F_{\lambda} \:=\:2x + 3y + 4z - 12 \:=\:0 \qquad \quad\;\;\; [4]



    Substitute [1], [2], [3] into [4]:

    . . 2\left(-\frac{1}{4}\lambda\right) + 3\left(-\frac{3}{2}\lambda\right) + 4\left(-\frac{2}{5}\lambda\right) -12\:=\: 0

    . . -\frac{66}{10}\lambda \:=\:12 \quad\Rightarrow\quad \lambda \:=\:-\frac{20}{11}



    Substitute into [1], [2], [3]:

    . . \begin{Bmatrix}x &=& \left(-\frac{1}{4}\right)\left(-\frac{20}{11}\right) &=& \dfrac{5}{11} \\ \\[-3mm]<br />
y &=& \left(-\frac{3}{2}\right)\left(-\frac{20}{11}\right) &=& \dfrac{30}{11} \\ \\[-3mm]<br />
z &=& \left(-\frac{2}{5}\right)\left(-\frac{20}{11}\right) &=& \dfrac{8}{11} \end{Bmatrix}

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