Let $f(x)$ defined on $[0, 1]$ be a smooth function with sufficiently many derivatives. $x_i = ih$, where $h =\frac{1}{N}$ and $i = 0,1,\cdots,N$ are uniformly distributed points in $[0, 1]$. What is the highest integer $k$ such that the numerical integration formula $$I_N=\frac{1}{N}(a_0(f(x_0)+f(x_N))+a_1(f(x_1)+f(x_{N-1}))+\sum_{i=2}^{N-2}f(x_i))$$ is $k$-th order accurate, namely $$\vert I_N-\int_{0}^{1}f(x)dx\vert\le Ch^k$$ for a constant $C$ independent of $h$? Please describe the procedure to obtain the two constants $a_0$ and $a_1$ for this $k$. $$$$ For this problem, one way is to use the Euler-Maclaurin formula and compare the coefficients to obtain the final result. My question is, what if I don’t know the Euler-Maclaurin formula? What’s the intuition for this problem? Are there any natural ways to come up with the solution? Thanks!
Edit1. We may apply the composite trapezoidal rule, but using composite trapezoidal rule will obtain 2nd derivative which (unfortunately) is not involving $f(x_1)$ and $f(x_{N-1})$, which is not what we want. I don’t know if the composite trapezoidal rule works, maybe this intuition is wrong