Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Continuous mathematics / 5 / b / Solution

By Taylor's theorem with Lagrange remainder, for an increment $t$,where ${\xi }_t$ lies between $x$ and $x+t$. The given third derivative bound implies $|R_x(t)|\le |t{|}^3/6$.