Solution
= Solution
Odd terms vanish and direct evaluation gives $L(p)=(8/45)c_4$, so the rule is exact through degree $3$. With $n=3$, the Peano kernel is
$$
K(t)=\frac16L[(x-t)_+^3]=\frac16\left[\frac{(1-t)^4}{4}-(-s-t)_+^3-(s-t)_+^3\right].
$$
Its nonnegativity and $\int K=L(x^4)/4!=1/135$ give $|L(f)|\le\|f^{(4)}\|_\infty/135$, so $N=135$. Equality holds for a quartic with fourth <derivative> of constant sign.