 Solution

The Chebyshev alternation theorem says that a polynomial $p$ of degree at most $n$ is a best uniform approximation to a real continuous function $f$ on a compact interval if and only if there are $n+2$ ordered points at which $f-p$ attains its uniform norm with alternating signs. For nonzero error these are equal-magnitude alternating extrema; the zero-error case is understood separately. The best approximating polynomial is unique.