Global minimum of RSS

Summary

Let

$$f\left( b_1,b_2 \right)=RSS=\sum_{i=1}^{n}{ e_i^2 }=\sum_{i=1}^{n}{ {\left( y_i-b_1-b_2x_i \right)}^2 }$$

Then $f\left( b_1,b_2 \right)$ is strictly convex .

The unique solution to the normal equations, given by the OLS formula is therefore the global minimum point of RSS.