= Solution
\b[The printed product omits the node with index zero.] The correct cardinal polynomial is
$$
L_k(x)=\prod_{\substack{0\le i\le n\\i\ne k}}\frac{x-x_i}{x_k-x_i}.
$$
For example, with two nodes zero and one, values zero and one, and both derivative data zero, the printed index range gives $L_1=1$ and its displayed interpolant is the constant one, which fails the value condition at zero. Thus the source formula cannot be proved literally.
With the corrected product, $L_k(x_j)=\delta_{kj}$. For $j\ne k$, $L_k^2$ has a double zero at $x_j$, so both <basis> functions and their derivatives vanish there. At $x_k$, put $\ell=L_k'(x_k)$. The value <basis> $H_k=L_k^2[1-2(x-x_k)\ell]$ satisfies $H_k(x_k)=1$ and $H_k'(x_k)=2\ell-2\ell=0$. The derivative <basis> $K_k=L_k^2(x-x_k)$ satisfies $K_k(x_k)=0$ and $K_k'(x_k)=1$. Both have degree at most $2n+1$. Consequently
$$
\boxed{h(x)=\sum_{k=0}^n(f_kH_k(x)+g_kK_k(x))}
$$
has every prescribed value and derivative, proving membership in the interpolation class after the required index correction. This corrected Hermite interpolant is used in the following parts.
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