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Math Help - LaGrange Multipliers

  1. #1
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    LaGrange Multipliers

    This was a question on my test yesterday and I was confused on how to do this. Could someone solve this one out for me?



    Find a point on the sphere x^2 + y^2 + z^2 = 4 that is the farthest away from point (1,-1,1)

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  2. #2
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    Hello,
    Quote Originally Posted by JonathanEyoon View Post
    This was a question on my test yesterday and I was confused on how to do this. Could someone solve this one out for me?



    Find a point on the sphere x^2 + y^2 + z^2 = 4 that is the farthest away from point (1,-1,1)

    Find an equation translating the "farthest away from point (1,-1,1)"

    That is you want to maximize the distance from a point on the sphere to this point.
    Let (x,y,z) \in S

    Let f ~:~ \mathbb{R}^3 \to \mathbb{R}
    f(x,y,z)=x^2+y^2+z^2-4

    Define \varphi ~:~ \mathbb{R}^3 \to \mathbb{R}
    \varphi(x,y,z)=\sqrt{(x-1)^2+(y+1)^2+(z-1)^2}

    f represents the constraint. That is if (x,y,z) \in S, f(x,y,z)=0


    Let \lambda be a Lagrange multiplier for this problem.
    Define h(x,y,z,\lambda)=f(x,y,z)+\lambda \varphi(x,y,z)

    Solve for \lambda in \nabla_{x,y,z,\lambda}  h(x,y,z,\lambda)=0, that is to say :

    \left\{\begin{array}{lllllll} \frac{\partial h}{\partial x}=0 \\ \\ \frac{\partial h}{\partial y}=0 \\ \\  \frac{\partial h}{\partial z}=0 \\ \\ \frac{\partial h}{\partial \lambda}=0 \end{array} \right.
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  3. #3
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    Uhhhmmm.. you lost me after the first line
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  4. #4
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    Quote Originally Posted by JonathanEyoon View Post
    Uhhhmmm.. you lost me after the first line
    If you don't even know the formula for distance (the first line), why are you even attempting a problem like this?
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