Least Squares Estimate (Derivation) | Simple Linear Regression

Deriving the least square estimates of β0 and β1 by setting them to values such that the sum of residuals squares is minimized. Setting the partial derivatives to 0 doesn't allow us to immediately conclude that the corresponding values of β0 and β1 minimizes the sum of residuals squares as the point we obtained could be a maximum or a saddle point. Of course, it is true that the point that we obtain does indeed result in a minimum, but I will not go through that justification and I'll simply focus on the conditions we get by setting the partial derivatives to 0.