why the text book method for finding the fitting curve is right ?
we have n data we want to approximate with a polynomial of degree m $P_m(x)$ (m < n-1).
and of course $E = \sum_{i=1}^m (y_i-P_m(x_i))^2$
to prove that the error function $E(a_0 , a_1 ,...,a_m)$ is Min where the partial derivatives of E with respect to $a_0 $ and $a_1$ , .. and $a_m$ are $0$; we must first show that this is the only critical point(call it c)
( if it's not the only critical point then we need to show E's value is Min in c when compared to other critical points ).
then we need to show that E's value in the boundary points are more than its value in c.
since we do not have a boundary (it's $\infty$) so we need to show when the function goes to $\infty$ in any direction the values of E are all more than c.
how can i prove this? why my textbook (burden) does not prove this? am i over-complicating the case?
$E$ is a coercive function, so has a global minimum (http://www.math.usm.edu/lambers/mat419/lecture4.pdf). The global minimum is a local minimum, so also a critical point. The only critical point is the global minimum.