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Math Help - Maximum and minimum

  1. #1
    Member roshanhero's Avatar
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    Maximum and minimum

    Find the minimum value of x^2+y^2+z^2 when x+y+z=3a^2.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by roshanhero View Post
    Find the minimum value of x^2+y^2+z^2 when x+y+z=3a^2.
    Apply Lagrange Multipliers.
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  3. #3
    MHF Contributor
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    Quote Originally Posted by roshanhero View Post
    Find the minimum value of x^2+y^2+z^2 when x+y+z=3a^2.
    The quickest way to do this is by geometry. The question is asking about the point in the plane x+y+z=3a^2 that is closest to the origin. More precisely, you want the square of this minimum distance. It's easy enough to see that the minimum occurs when x=y=z=a^2. So the minimum value of x^2+y^2+z^2 is 3a^4.
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  4. #4
    Member roshanhero's Avatar
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    Can you tell me how to solve it by using lagrange multipliers.
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  5. #5
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    Quote Originally Posted by roshanhero View Post
    Can you tell me how to solve it by using lagrange multipliers.
    Construct L = x^2 + y^2 + z^2 + \lambda (x + y + z - 3a^2).

    Calculate the partial derivatives with respect to x, y, z and \lambda and solve the following four equations simultaneously:

    \frac{\partial L}{\partial x} = 0 .... (1)

    \frac{\partial L}{\partial y} = 0 .... (2)

    \frac{\partial L}{\partial z} = 0 .... (3)

    \frac{\partial L}{\partial \lambda} = 0 \Rightarrow x + y + z = 3a^2 .... (4)
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