Solution
= Solution
Unbiasedness requires $\sum c_iX_i=1$. Cauchy–Schwarz gives $1\le(\sum c_i^2)S_{xx}$, so
$$
\operatorname{Var}(\tilde\beta)=\sigma^2\sum c_i^2\ge\frac{\sigma^2}{S_{xx}}.
$$
Equality holds exactly for $c_i=X_i/S_{xx}$.
= Solution
Unbiasedness requires $\sum c_iX_i=1$. Cauchy–Schwarz gives $1\le(\sum c_i^2)S_{xx}$, so
$$
\operatorname{Var}(\tilde\beta)=\sigma^2\sum c_i^2\ge\frac{\sigma^2}{S_{xx}}.
$$
Equality holds exactly for $c_i=X_i/S_{xx}$.