The following problem was given in my math book:
I can't see what's wrong with the analysis for the life of me. I suspect it has something to do with $f'''(\xi_1)-f'''(\xi_2)$ being greater than or equal to h, perhaps? But I just cannot wrap my head around what could be wrong here. Thanks in advance for the help!
It's not that the analysis is wrong, it's just that it gives an unnecessarily bad result. What happens if you go one step further in the expansions before adding them up? (Of course under the assumption that $f$ is nice enough to allow this.)