proof that the sum of the squares achieves minimum

I'm analyzing the Least Squares Fitting algorithm on this site.

You can read there:
The condition for $R^2$ to be a minimum is that $$\frac{\partial (R^2)}{ \partial a_i} = 0$$

But I learned that this condition stands for extremum. Minimum or maximum!

How to proof that $R^2$ (the sum of squares) achieves there it's minimum?

I'm interested in a simple two dimensional proof.

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Do you believe that $R^2$ has a minimizer? If so, this minimizer must satisfy the equation you wrote down. And if that equation has only one solution, then you have found your minimizer. – littleO Oct 28 '12 at 20:06

1 Answer

Well, are there any curves with a maximum distance from the points?

How would such curve look like?

(Hint: There is no such curve, if we suppose we have a curve with maximal distance from all points, then we may nudge a bit "away" from the points, and increase distance).

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