Sharp one-panel trapezoidal error bound ($|I(f)-T(f)|\le \frac{(b-a{)}^3}{12}{\max }_{[a,b]}|f^{\prime \prime }|$)

For $f\in C^2[a,b]$ with $a<b$, the trapezoidal rule satisfies the displayed bound, and the constant $1/12$ is best possible. The polynomial interpolation error for the endpoint linear interpolant satisfies $|f(x)-p(x)|\le \frac{1}{2}M(x-a)(b-x)$, where $M=\max |f^{\prime \prime }|$. Integrating this inequality and using the triangle inequality for an integral proves the bound because ${\int }_a^b(x-a)(b-x)dx=(b-a{)}^3/6$. Equality holds for $f(x)=(x-a{)}^2$: the exact integral is $(b-a{)}^3/3$, its trapezoidal rule estimate is $(b-a{)}^3/2$, and $M=2$.