# Math Help - [SOLVED] Higher Order Interpolation

1. ## [SOLVED] Higher Order Interpolation

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

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

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

2. Well, the right method to use was Hermite interpolation, but my textbook never specified that it could go beyond $K(x_i) = f(x_i)\text{ and } K'(x_i) = f'(x_i)$...