Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / P / 2

Let $M_j=s^{\prime \prime }(j)$. A natural cubic spline has $M_0=M_2=0$. The interior spline equation on equal unit intervals isso $M_1=6$. The standard second-derivative interpolation formula, or direct integration of the linear second derivative, givesThe two pieces have equal value, first derivative, and second derivative at one, and $s^{\prime \prime }$ vanishes at zero and two.

Adding the linear function $x$ changes the nodal data by $(0,1,2)$ without affecting second derivatives. By uniqueness, the second requested natural cubic spline is $s(x)+x$, with pieces $x^3$ and $-x^3+6x^2-6x+2$ respectively.