# Composite midpoint and trapezoid quadrature of twice differentiable function

This is a homework problem that I am having trouble with.

Given $f: \mathbb{R} \to \mathbb{R}$ is twice differentiable ans $f''(x) \geq 0$ on the interval $x \in [a,b]$. Prove that: $$M_k(f) \leq \int_a^b f(x) \mathrm{d}x \leq T_k(f)$$

Where $M_k(f)$ is the composite Midpoint rule and $T_k(f)$ is the composite trapezoid rule.

I can see this conceptually. $f$ is convex so the whole function lies under a secant line through $a$ and $b$ which indicates $\int_a^b f(x) \mathrm{d}x \leq T_k(f)$, and I think this also indicates the left part. But I have a hard time proving this.

Any tips to get me in the right direction?

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Have you tried proving for the case of a single interval? – J. M. Oct 16 '11 at 17:31
Yes. I tried a Taylor expansion on a single interval which shows that the area I(f) consists of M(f) + a second order derivitive term + error. Which indicates the left part if the error is zero. Im not sure what to do with the error term tho. – Zagga Oct 16 '11 at 17:36
Has the Euler-Maclaurin formula ever been mentioned in your class, at least in passing? – J. M. Oct 16 '11 at 17:43
No it has not been mentioned. – Zagga Oct 16 '11 at 17:45