= Sharp Peano bound for an interior three-point first derivative
{title2=$|L(f)|\le\|f'''\|_\infty/36$}
For $L(f)=-5f(0)/3+4f(1/2)/3+f(1)/3-f'(1/3)$ on $C^3[0,1]$, quadratic polynomials are annihilated. Taylor's formula with integral remainder gives $L(f)=\int_0^1K(t)f'''(t)dt$ 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.
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