I was asked to use Hermite Interpolation to find an interpolating polynomial of a degree at most $5$ that satisfies the conditions:
| $x = 0$ | $x = 1$ | $ x= 3$ | |
|---|---|---|---|
| $f(x) = $ | $1$ | $3$ | $1$ |
| $f'(x) = $ | $0$ | $3$ | |
| $f''(x) = $ | $6$ |
And to find the value of a first derivative of that polynomial at $x = 4$ (described below as $b_2$) without calculating the whole derivative f'(x).
Now I know how how to do the first part. It's the second one that bothers me (bolded). In a hint I was told to write 2 values of $x = 4$ 'at the bottom of the pyramid' (see the scheme below) and to use backward reasoning (from top of the pyramid to the bottom) since I know that the degree of the polynomial is at most 5 (so at the very top of the pyramid there will be only zeros). I got a scheme show below (by Hermite Interpolation, with notation as in Wikipedia):
$$\begin{array}{lllll} x = 0 & f(x) = 1\\ &&0\\ x = 0 & f(x) = 1&&3\\ &&0&&-1\\ x = 0 & f(x) = 1&&2&&0\\ &&2&&-1&&0\\ x = 1 & f(x) = 3&&1&&0\\ &&3&&-1&&a_6\\ x = 1 & f(x) = 3&&-2&&a_5\\ &&-1&&a_4&&b_6\\ x = 3 & f(x) = 1&&a_3&&b_5\\ --- &&a_2&&b_4\\ x = 4 & f(x) = a_1&&b_3\\ &&b_2\\ x = 4 & f(x) = b_1\\ \end{array}$$
I know that: $$0 = \frac{a_6 - 0}{4-0}$$ $$0 = \frac{b_6 - a_6}{4-0}$$ and so on.
And it works! I got $b_2 = -24$. It is the correct answer (I know by calculating the whole derivative). But my question is how? How does the algorithm work if I didn't 'provide an information that I want to calculate the value of first derivative at $x = 4$' let's say. So why is it calculated at $x = 4$ (and not e. g. at $x = 10$)?
Additional information: The polynomial is: $1+3x^2 -x^3$, derivative is: $6x-3x^2$