We assume that $I = [0,1]$, $\forall x \in I, w(x) = 1$, and we consider the quadrature formula
$$I_1(g) = \frac{g(0) + g(1)}{2},$$
which is of order $m = 1$.
Calculate the associated Peano kernel $t \mapsto K_1(t)$ and show that, for every function $g$ of class $\mathcal{C}^2$ from $[0,1]$ to $\mathbb{R}$, we have the following bound on the associated quadrature error:
$$|e(g)| \leqslant \frac{1}{12} \sup_{x \in [0,1]} |g''(x)|.$$