 Solution

The remainder in part (b) involves an unspecified $\xi =\xi (x)$. Its existence for every $x$ does not ensure a differentiable selection. Even if such a selection existed and $f$ had an additional derivative, the product rule and chain rule would giveThe first term cannot be discarded or bounded from the given information. Neither $f^{(n+2)}$ nor ${\xi }^{\prime }$ is supplied by the original theorem.

For an explicit demonstration, interpolate the cubic polynomial $f(x)=x^3$ at zero and one. Then $p_1(x)=x$, andDifferentiating correctly givesTreating the remainder coefficient $x+1$ as a constant would omit $x(x-1)$ and give a wrong answer at every interior point. An existential intermediate point cannot be frozen when differentiating. Part (i) instead applies Rolle's theorem directly to the polynomial interpolation error.