1. ## 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.

2. ## Re: Absolute Minimum

Originally Posted by brucewayne
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.