Sharp Peano bound for an interior three-point first derivative ($|L(f)|\le \Vert f^{\prime \prime \prime }{\Vert }_{\infty }/36$)

For $L(f)=-5f(0)/3+4f(1/2)/3+f(1)/3-f^{\prime }(1/3)$ on $C^3[0,1]$, quadratic polynomials are annihilated. Taylor's formula with integral remainder gives $L(f)={\int }_0^{1}K(t)f^{\prime \prime \prime }(t)dt$ with $K(t)=2(1/2-t{)}_{+}^2/3+(1-t{)}^2/6-(1/3-t{)}_{+}$. This Peano kernel is nonnegative throughout the interval and integrates to $1/36$, giving the sharp bound; $f(x)=x^3/6$ attains equality. The derivative is at an interior point, so this is distinct from an endpoint differentiation formula.