I have this question in my homework in Numerical Analysis but I can't figure it out. Can someone have a look and help me?
Question:
Let $E_0(f)$ and $E_1(f)$ be the quadrature errors of the midpoint and the trapezoid formula, respectively. Prove that $E_1(f) \approx 2|E_0(f)|$.
$\textbf{Comments:}$
Let's say we are integrating function $f$ in the interval $[a,b]$. The quadrature error for the midpoint formula (as far as I know) is $E_0(f)=\frac{(b-a)^3}{24}f''(\xi)$ where $\xi \in (a,b)$ and the quadrature error for the trapezoid formula is $E_1(f)=-\frac{(b-a)^3}{12}f''(\xi)$ where $\xi \in (a,b)$. I don't see why $f''(\xi)$ is negative so that I can prove the requested statement. Any hint?
