= Solution
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 $J(R)\geq J(S)$, where $J(u)=\int_{x_0}^{x_n}(u'')^2\,dx$. Applying the same argument with their roles exchanged gives $J(S)\geq J(R)$. Hence equality holds, and the exact identity from part (i) yields
$$
\int_{x_0}^{x_n}(R''-S'')^2\,dx=0.
$$
The <integrand> is continuous and nonnegative. If it were positive somewhere, it would have a positive <integral> over a small <interval>. It follows that $R''=S''$ 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.
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