OLS estimator derivation: second-order condition to prove global minimum?
It seems that you working on the right lines. By multiplying out the brackets, it can easily be verified that $n\sum{X}_i^2 - (\sum{X}_i)^2 = n\sum{(X_i - \bar{X}})^2$ where $\bar{X}$ is the mean. As you yourself spotted, this is basically a standard expression for the variance. From this, it follows that $n\sum{X}_i^2 - (\sum{X}_i)^2 &gt; 0$ provided that $X_i \neq \bar{X}$ for some $i$.

Incidentally, you also need to verify that one of the second derivatives is positive (say $SSR{_{\alpha\alpha}}&gt;0$) which follows directly from your Hessian.