  Given query access to a monotone function $f\colon\{0,1\}^n\to\{0,1\}$ with
certificate complexity $C(f)$ and an input $x^{\star}$, we design an algorithm
that outputs a size-$C(f)$ subset of $x^{\star}$ certifying the value of
$f(x^{\star})$. Our algorithm makes $O(C(f) \cdot \log n)$ queries to $f$,
which matches the information-theoretic lower bound for this problem and
resolves the concrete open question posed in the STOC '22 paper of Blanc, Koch,
Lange, and Tan [BKLT22].
  We extend this result to an algorithm that finds a size-$2C(f)$ certificate
for a real-valued monotone function with $O(C(f) \cdot \log n)$ queries. We
also complement our algorithms with a hardness result, in which we show that
finding the shortest possible certificate in $x^{\star}$ may require
$\Omega\left(\binom{n}{C(f)}\right)$ queries in the worst case.
