Past exam of the numerical analysis course of the University of Oxford / 2017 / 3 / b / ii

Let $R$ be a second cubic spline matching the same values and endpoint derivatives. The minimum roughness property of a clamped cubic spline applied with $S$ as the minimizing cubic spline gives $J(R)\ge J(S)$, where $J(u)={\int }_{x_0}^{x_n}(u^{\prime \prime }{)}^{2}dx$. Applying the same argument with their roles exchanged gives $J(S)\ge J(R)$. Hence equality holds, and the exact identity from part (i) yieldsThe integrand is continuous and nonnegative. If it were positive somewhere, it would have a positive integral over a small interval. It follows that $R^{\prime \prime }=S^{\prime \prime }$ everywhere. Consequently $R-S$ is an affine function. Its value and derivative vanish at $x_0$, so it is identically zero. Thus $R=S$, proving uniqueness of the clamped cubic spline with the prescribed data. Existence was assumed in the question; this argument establishes the remaining uniqueness claim.