Let \(p\) be a polynomial \(p\) of degree \(m\) on \([0,1]\). Given \(a \in \{0, 1\}\), the authors demonstrate in a constructive way that there exists a sequence \((p_n)\) of polynomials of degree \(\max(m,n+1)\) such that, for all \(n\), (i) \(p_n\) interpolates \(p\) at the points \(k/n\), \(k = 0, 1, 2, \ldots, n\); (ii) \(p_n'(a) = 0\); and (iii) \(\| p_n - p \|_\infty \le |p'(a)| / (4n)\) (and hence \(p_n \to p\) uniformly). An analog question is then investigated for \(a = 1/2\). In this case, it turns out that the interpolating polynomials that satisfy (i) and (ii) exhibit a Runge phenomenon, and so the sequence \((p_n)\) becomes unbounded and hence divergent. This can, however, be repaired for \(a = 1/2\) and odd \(n\) if one replaces the equidistant interpolation points in (i) by zeros of the Chebyshev polynomials of the first kind (translated to the interval \([0,1]\)), in which case an analog of (iii) holds.