My goal is to find an (exact) error expression for Richardson extrapolation applied to the composite midpoint rule. I know that the error for this rule is $$\displaystyle\frac{(b-a)h^2}{6}f^{''}(\xi)$$ and the Richardson extrapolation in this case would be equal to $$I_h^R(f)(x=\displaystyle\frac{I_h(f)(x)-4I_{h/2}(f)(x)+3f(x)}{-3}$$ where $I_h$ is the composite midpoint rule with step $h$ and so on. When substituting looking for the error, I reach an expression of the form $$\displaystyle\frac{(b-a)h^2(f^{''}(\xi)-f^{''}(\xi'))}{-18}$$ I thought of applying MVT, but it's unknown if those two points are less than a distance of $h$ apart.
Any help is appreciated.