 Solution

Put $E=T-S$. Matching interpolation data and endpoint derivatives givesThe endpoint slope constraints make $S$ a clamped cubic spline. The decisive step in its minimum roughness property of a clamped cubic spline is the orthogonality identityProve it separately on each interval. Two applications of integration by parts, using the vanishing fourth derivative of the cubic piece, giveThe second boundary term is zero because $E$ vanishes at both endpoints of this interval. Sum over $i$. At each interior spline knot, the two values of $S^{\prime \prime }{E}^{\prime }$ agree because both factors are continuous, so the contributions cancel. The remaining outer boundary terms vanish because $E^{\prime }$ is zero at both outer endpoints. This proves the identity.

Expanding the square now gives the stronger exact equalityThe last integral is nonnegative, proving the required inequality. It measures the excess second derivative roughness penalty of $T$ over $S$.

The argument applies directly to twice continuously differentiable competitors. More generally it holds when $T^{\prime }$ is an absolutely continuous function and $T^{\prime \prime }$ is square-integrable, the usual finite-energy class for this second derivative roughness penalty. Mere existence of pointwise second derivatives does not by itself guarantee that all displayed integrals exist; the variational statement is understood for admissible competitors with this regularity.