 Strict decrease of best polynomial approximation under a derivative sign

For real $f\in C^{n+1}[0,1]$ with positive $(n+1)$st derivative, equality of consecutive best uniform approximation errors would make a best degree-at-most-$n$ polynomial also best at degree $n+1$. The Chebyshev alternation theorem then supplies $n+3$ alternating extrema of its nonzero error, hence $n+2$ distinct zeros. Repeated Rolle's theorem forces a zero of the $(n+1)$st derivative of that error, contradicting the assumed sign. The same conclusion holds for a strictly negative derivative by replacing $f$ with $-f$.