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on Optimal symmetric three-point quadrature for a bounded second derivative

The statement

For $0\le a\le1/2$ , define $Q_a(f)=(f(a)+f(1/2)+f(1-a))/3$ . Let $C(a)=\sup_{\|f''\|_\infty\le1}|\int_0^1 f(x)\,dx-Q_a(f)|$ , where the supremum is over twice weakly differentiable real functions. Determine the minimizing node $a_*$ and the sharp value $C(a_*)$ .

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