 Solution

It suffices to assume $f\in C^2[a,b]$. The polynomial interpolation error for the endpoint linear interpolant gives, for every interior $x$, some ${\xi }_x\in (a,b)$ withLet $M={\max }_{[a,b]}|f^{\prime \prime }|$, which exists by continuity of the second derivative and the extreme value theorem. Since $(x-a)(b-x)\ge 0$,Integrating this pointwise inequality and using the triangle inequality for an integral yieldsThusThis derivation does not require choosing ${\xi }_x$ as a continuous function of $x$; the pointwise upper bound is sufficient.

The constant is sharp. Take the quadratic polynomial $f(x)=(x-a{)}^2$ and write $L=b-a$. Its exact integral is $L^3/3$, its trapezoidal rule value is $L^3/2$, and $\max |f^{\prime \prime }|=2$. ConsequentlyEquality holds, so no smaller universal constant than $1/12$ works under these assumptions. This is the sharp one-panel trapezoidal error bound.