## sunsunsun1225 3 years ago Hi can anyone show how to use second derivative test to prove that the deviation of linear regression has a local minimum but not a maximum? thanks

The second derivative test $f_{aa} f_{bb} - f_{ab}^2$reduces to 4n^2( VAR(X)) where VAR(x) = $1/n \sum_{i=1}^{n} (x_i - \mu)^2$ and $\mu = 1/n \sum_{i=1}^n x_i$. This is always positive, hence not a saddle point. $f_{aa} = 2 \sum_{i=1}^n x_i^2$ which is always positive, hence a local minimum.