Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 318 / 2 / a / Solution

The Chebyshev alternation theorem says that $p^{\ast }\in {\mathcal{P}}_n$ is the unique best uniform approximation to $f\in C[-1,1]$ if and only if there are at least $n+2$ ordered pointsand a sign $\epsilon \in \{-1,1\}$ such thatThus the extremal error has common magnitude and alternating sign at $n+2$ points.