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There exist well-known tight bounds on the error between a function $f \inC^{\,n + 1}([-1, 1])$ and its best polynomial approximation of degree $n$. Weshow that the error meets these bounds when and only when $f$ is a polynomialof degree $n + 1$.

The degree-$(n+1)$ polynomials are the most difficult $C^{\,n + 1}$ functions to uniformly approximate with degree-$n$ polynomials