= Sharp Peano-kernel constant for the three-point endpoint first derivative
{title2=$|L(f)|\le\frac13\|f^{(3)}\|_\infty$}
For $L(f)=f'(0)+\frac32f(0)-2f(1)+\frac12f(2)$ on $C^3[0,2]$, the <Peano kernel> is
$$
K(t)=\begin{cases}t-\frac34t^2,&0\le t\le1,\\\frac14(2-t)^2,&1\le t\le2.\end{cases}
$$
The <Taylor expansion> with <integral> remainder gives $L(f)=\int_0^2K(t)f^{(3)}(t)\,dt$, because $L$ annihilates <polynomials> of degree at most two. Since $K\ge0$ and $\int_0^2K=1/3$, the sharp bound is $|L(f)|\le\frac13\|f^{(3)}\|_\infty$. Equality holds for $f(x)=x^3/6$.
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