We are looking for the solution of the Hermitian-polynomial-interpolation with the data points $x_0=x_1=0, x_2=2, x_3=x_4=1.$
Construct, analogous to the Lagrange-interpolation, basis functions $L_0,\ldots,L_4 \in \Pi_4$ such that $\mu_j(L_k)=\delta_{jk}$ for $j,k=0,\ldots,4.$
We defined $d_j=\max\{l:x_j=x_{j-l}\}$ and $\mu_j(f):=f^{(d_j)}(x_j)$. And the Lagrange basis polynomials are defined as $L_j(x):=\prod_{k=0,k\neq j}^{n}\frac{x-x_k}{x_j-x_k}$.
So $d_0=0,d_1=1,d_2=0,d_3=0,d_4=1$.
What do I have to do now? Determine $L_k$ and see if they satisfy $\mu_j(L_k)=\delta_{jk}$? And what do I have to do if they dont satisfy this? I don't really understand the task.