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\noindent {\bf Problem statement} Given $n$ data points
$(x_1,y_1),\ldots,(x_n,y_n)$, we may seek a linear function $y=mx+b$
that best fits the data. The {\bf linear least-squares fit} is the
linear function $f(x)=mx+b$ that minimizes the sum of the squares (see
the Figure) $\displaystyle E(m,b)=\sum_{nj=1}^n (y_j-f(x_j))^2$.

\noindent Show that $E$ is minimized for $m$ and $b$ satisfying
\smallskip \centerline{$\displaystyle m\sum_{j=1}^n
x_j+bn=\sum_{j=1}^n y_j$ and $\displaystyle m\sum_{j=1}^n
x_j^2+b\sum_{j=1}^n x_j=\sum_{j=1}^n x_j y_j$}

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\vtop{\hsize=2.55in \noindent {\bf Comment} This is problem 44 in the
textbook's section 14.7.  The result is quite important in practical
computation. Several assertions must be verified: that $E$ has one
critical point which is a local minimum, and that this local minimum
is actually an {\it absolute}\/ minimum.

}

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