Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / A / 3 / c / Solution

The midpoint error annihilates affine functions, so here $n=1$. Apply the functional to $(x-t{)}_{+}$:ThusIt is nonnegative, and symmetry gives ${\int }_0^{h}K=2{\int }_0^{h/2}{t}^2/2dt=h^3/24$. ThereforeFor a quadratic with constant nonzero second derivative equality holds, so this midpoint quadrature rule error constant is sharp.