Mathematics

The total squared distance is a smooth function of your parameter—think of it as a landscape with hills and valleys. At any minimum point, the landscape must be flat (zero slope), because if the slope were nonzero, you could move slightly along the parameter to reduce the distance further. The second derivative being positive confirms you've found a valley, not a peak or saddle point. This is just calculus: optimizers occur where $\frac{d}{d\theta}(\text{total squared distance}) = 0$, and the second derivative test distinguishes minima from maxima.

The total squared distance is a smooth function of your parameter—think of it as a landscape with hills and valleys. At any minimum point, the landscape must be flat (zero slope), because if the slope were nonzero, you could move slightly along the parameter to reduce the distance further. The second derivative being positive confirms you've found a valley, not a peak or saddle point. This is just calculus: optimizers occur where $\frac{d}{d\theta}(\text{total squared distance}) = 0$, and the second derivative test distinguishes minima from maxima.