I've been going over this problem again and again and I can't seem to figure out what interpolation method to use.

If $\displaystyle f \in C^2[a,b]$ and $\displaystyle a = x_0 < x_1 < \cdots < x_n = b$, and the following values are given:

$\displaystyle f(x_0)\quad f(x_1)\: \cdots \: f(x_n)$

$\displaystyle f'(x_0)\quad f'(x_1)\: \cdots \: f'(x_n)$

$\displaystyle f''(x_0)\quad f''(x_1)\: \cdots \: f''(x_n)$

Give a formula for the (unique) polynomial $\displaystyle K_{3n+2}(x)$ of degree $\displaystyle \leq 3n+2$ where $\displaystyle K_{3n+2}(x_j) = f(x_j)$, $\displaystyle K'_{3n+2}(x_j) = f'(x_j)$ and $\displaystyle K''_{3n+2}(x_j) = f''(x_j)$ for $\displaystyle j=0,\ldots,n$

I'd really appreciate any help you could offer. Thanks a lot!