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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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    No it has not been mentioned.2011-10-16

0 Answers 0