I'm bit confused about finding critical points of functions.
Studying the least square method we got some data $\{(y_{1},x_{1}),...,(y_{n},x_{n}))\}$ and can define error associating with $y=ax+b$ by
$$E(a,b)=\displaystyle\sum_{i=1}^{n}(y_{i}-[ax_{i}+b])^{2}$$
We want to find the minimum of $E$ and so we find where its gradient is equal to zero. i.e.
$$\frac{\partial E}{\partial a}=0=\frac{\partial E}{\partial b}$$
but, by doing that we would fint just a critical point, it could be a minimum or maximum or neither of them.
But anyway this calculus give me the exactly minimum points, my doubt is why that? do we have the second derivative criterion here?