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Math Help - Absolute Minimum

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    Absolute Minimum

    Here is the function: f(x,y,z)=(x-y)^2+(y-z)^2+(x-z)^2=0. How do I verify that x=y=z=0 is the absolute minimum and there are no others? The function above stems from another function in which I found the partial derivative and completed the square yielding the above result.
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  2. #2
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    Re: Absolute Minimum

    Quote Originally Posted by brucewayne View Post
    Here is the function: f(x,y,z)=(x-y)^2+(y-z)^2+(x-z)^2=0. How do I verify that x=y=z=0 is the absolute minimum and there are no others? The function above stems from another function in which I found the partial derivative and completed the square yielding the above result.
    For an absolute minimum we have

    \displaystyle \begin{align*} \nabla f &= \mathbf{0} \\ \left( \frac{\partial f}{\partial x} ,  \frac{\partial f}{\partial y} , \frac{\partial f}{\partial z} \right) &= \left( 0, 0, 0 \right) \\ \left( 2(x - y) + 2(x - z) , -2(x - y) + 2(y - z) , -2(y - z) -2(x - z) \right) &= \left( 0,0,0 \right) \end{align*}

    Now equate each component and try to solve the three equations simultaneously for x, y, z.
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