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Math Help - Finding max value of a 3-variable function

  1. #1
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    Finding max value of a 3-variable function

    Find the maximum value of f(x,y,z) = x^2 + xy + y^2 + yz + z^2 on the unit sphere.

    I don't know whether I should parametrize the unit sphere and use a composition of functions, or go straight into Lagrange multipliers. Either way, I don't know how to solve the resulting equations.
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  2. #2
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    Quote Originally Posted by tjkubo View Post
    Find the maximum value of f(x,y,z) = x^2 + xy + y^2 + yz + z^2 on the unit sphere.

    I don't know whether I should parametrize the unit sphere and use a composition of functions, or go straight into Lagrange multipliers. Either way, I don't know how to solve the resulting equations.
    Set g(x, y, z) = x^2 + y^2 + z^2. You want to solve grad(f) = lambda*grad(g), g = 1. (It may be simpler to use lambda*grad(f) = grad(g) in this case.) If you can't solve the equations, please state where you're having difficulty.
    Last edited by rn443; October 28th 2009 at 01:00 AM.
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  3. #3
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    So you have to solve the following four equations
    (1) 2x + y = 2λx
    (2) x + 2y + z = 2λy
    (3) y + 2z = 2λz
    (4) x^2 + y^2 + z^2 = 1

    Is there a usual way to solve a system like this?

    What I did was solve (1) for y and plugged that y into (3). That gave me λ=1, which then gave me y=0, x = sqr(1/2), and z = ∓ sqr(1/2). However, this does not give the maximum value of f. Does this just mean that there are other points (x,y,z,λ) that satisfy these equations? How do you find all of them?
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  4. #4
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    Quote Originally Posted by tjkubo View Post
    So you have to solve the following four equations
    (1) 2x + y = 2λx
    (2) x + 2y + z = 2λy
    (3) y + 2z = 2λz
    (4) x^2 + y^2 + z^2 = 1

    Is there a usual way to solve a system like this?

    What I did was solve (1) for y and plugged that y into (3). That gave me λ=1, which then gave me y=0, x = sqr(1/2), and z = ∓ sqr(1/2). However, this does not give the maximum value of f. Does this just mean that there are other points (x,y,z,λ) that satisfy these equations? How do you find all of them?
    The maximum of f under the constraint can be obtained by plugging the points you found into the equation for f and finding which is bigger.
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  5. #5
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    Yes, but the trouble is finding those points.
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  6. #6
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    Quote Originally Posted by tjkubo View Post
    Yes, but the trouble is finding those points.
    You're forgetting the case where x = z. That's why you're dividing by (x - z) to get lambda = 1.
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