Solution
= Solution
<Taylor's theorem> at zero gives $f(x)=-1000+x+\tfrac12x^2f''(\xi)$ 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 gives
$$
f(2000)>-1000+2000-\frac{2000^2}{8000}=500>0.
$$
By the <intermediate value theorem> there is a root strictly between these values. The given uniqueness identifies it as \b[$1000<x^*<2000$].