 Solution

Use the endpoint continuity of $f$ from part b, and interpret the displayed endpoint derivatives as one-sided endpoint values. Starting with $f(a)=f(b)=0$, Rolle's theorem gives $c_1\in (a,b)$ with $f^{\prime }(c_1)=0$. Since also $f^{\prime }(b)=0$, apply Rolle's theorem to $f^{\prime }$ on $[c_1,b]$ to obtain $c_2\in (c_1,b)$ with $f^{\prime \prime }(c_2)=0$. Repeating givesFor $j<n$, the lower derivative $f^{(j)}$ is continuous in the interior by differentiability implies continuity, and at $b$ by the existence of its next one-sided derivative. At the final step, the stipulated continuity of $f^{(n)}$ on $[a,b]$ and its differentiability on $(a,b)$ give exactly the hypotheses of Rolle's theorem on $[c_n,b]$. ThereforeThe endpoint regularity matters: merely assigning arbitrary values to $f$ at $a,b$ while prescribing only the interior derivatives would not justify the first application of Rolle's theorem.