= Solution
{c}
Differentiate the <ordinary least squares> objective in <7/a>. Its <normal equations for linear least squares> are
$$
\sum_i(Y_i-\widehat\alpha-\widehat\beta x_i)=0,\qquad
\sum_i x_i(Y_i-\widehat\alpha-\widehat\beta x_i)=0.
$$
Using $\sum_i x_i=0$ and $S_{xx}=\sum_i x_i^2>0$ gives
$$
\boxed{\widehat\alpha=\overline Y,\qquad
\widehat\beta=\frac{\sum_i x_iY_i}{S_{xx}}.}
$$
The <Hessian matrix> of the objective in $(a,b)$ is $2\operatorname{diag}(n,S_{xx})$, which is <positive-definite>, so this is its unique <global minimum>. For the observed <estimates>, replace $Y_i$ by $y_i$.
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