Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 4 / b / Solution

The second-order remainder form of Taylor's theorem gives, for a point ${\xi }_x$ between $0$ and $x$,For $x\ne 0$, the given curvature bounds implyIn particular $f(1)<-1/2<0$ and $f(2)\ge 5>0$. Differentiability implies continuity, so the intermediate value theorem gives a root in $(1,2)$.

The mean value theorem applied to $f^{\prime }$ gives $f^{\prime }(x)=3+x{f}^{\prime \prime }({\eta }_x)\ge 3+2x>0$ for $x\in (1,2)$. Thus $f$ is strictly increasing there, and it cannot have two roots. There is exactly one root in $(1,2)$.