Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 6 / b / Solution

Taylor's theorem at zero gives $f(x)=-1000+x+\frac{1}{2}{x}^2{f}^{\prime \prime }(\xi )$ for a point $\xi$ between zero and $x$. At $x=1000$, strict negativity of the second derivative gives $f(1000)<0$. At $x=2000$, its lower bound givesBy the intermediate value theorem there is a root strictly between these values. The given uniqueness identifies it as $1000<x^{\ast }<2000$.