Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 4 / c / ii

On the panel $[2j/n,2(j+1)/n]$, the maximum fourth derivative is at the left endpoint because it is decreasing. ThusThe geometric series and the sum of consecutive integers giveSubstitution provesFor $n>2$, at least one panel's maximum is smaller than the global maximum $1$, so this bound is strictly tighter than the global bound. For $n=2$, the two bounds coincide.