I have question about quadratic spline interpolation.

To develop S_{2,N}(x), the piecewise quadratic "N" splines for given data (x_i, y_i) for i = 0,1,... N , why one have to assume first derivative of S_{2N}(x) is zero at x = xo, that is S_{2,N}(xo) = 0.

I mean, why that deritative is zero?

Some text books force second derivative of first spline, i.e. S_{2,1}(x) is zero at x = xo.

Are both conditions equivalent? How ?