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

The printed inequality has its sign reversed: $f(x)<f_2(x)$ for $x>1$. This is present in the original scan, rather than arising in transcription. For $x>1$, Taylor's theorem with Lagrange remainder givesPart (a) has $f^{\prime \prime \prime }(\xi )<0$, so the remainder is strictly negative. As an independent check, direct algebra gives the quadratic Taylor majorant of a reciprocal quadratic:For example, $f(2)=1/5<1/4=f_2(2)$ is already a counterexample to the requested direction. The exact identity also shows that this polynomial majorizes $f$ on the entire real line, with equality only at one.