Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 6 / f

For $n$ equal strips and a uniform bound $|f^{\prime \prime }(x)|\le M$ on $[a,b]$, the composite trapezoidal error bound isOn $[1,25]$, part (b) gives $M=1/2$: $f^{\prime \prime }$ is positive and decreases from $1/2$ after $x=1$. HenceTo certify error at most $10^{-4}$, choose $n\ge 2400$; equality gives exactly $10^{-4}$ in this bound. This is a sufficient certificate, not a claim that every smaller $n$ fails in actual numerical error.

Strictly, the maximum specified over the open interval $(1,25)$ is not attained; its supremum is $1/2$. Using the maximum on the closed interval, or simply the uniform bound $M=1/2$, gives the valid error estimate above.