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

Four differentiations giveFor $0<x<1$, the fifth derivative is negative, so the fourth derivative is strictly decreasing. Since $f^{(4)}(1)=0$, strict decrease towards that endpoint implies $f^{(4)}(x)>0$ for $x<1$. On the closed interval, $0\le f^{(4)}(x)\le 1$, with the maximum at $0$ and minimum at $1$.