## Interpolation

suppose that $n \geq 1$. The function $f$ and its derivatives of order up to and including $2n+1$ are continuous on [a,b]. The points $x_i, i = 0,1,...n$ are distinct and lie in [a,b]. Construct polynomials $r_0 (x), h_i (x), k_i(x) , i= 1,...,n$ of degree $2n$ such that the polynomial $p_{2n} (x) = r_0 (x)f(x_0) + \sum_{i=1}^n h_i (x)f(x_i) + k_i (x)f ' (x_i)$ satisfies conditions $p_{2n} (x_i) = f(x_i), i=0,1,...n$ and $p ' _{2n} (x_i) = f ' (x_i), i=1,....n$

I can't get anywhere with this. My professor doesn't teach examples, only theorems... help is much appreciated =)