I'm trying to show the existence and uniqueness of an interpolating polynomial $p$ of degree less than or equal to 3 that interpolates a differentiable function $f$ ( i.e. $f(x_i) = p(x_i)$ for $i = 0, 1, 2 $ ) and it is also given that that $f'(x_1) = p'(x_1)$.
The way I'm approaching the problem is trying a polynomial of the type:
$ p(x) = a_0 +a_1x + a_2x^2+a_3x^3 $ and solving for $a_0, a_1, a_2, a_3$. In this way, I have to show that the determinant of the matrix:
$ A = \begin{bmatrix}1& x_0 & x_0^2 &x_0^3\\1& x_1 & x_1^2 &x_1^3\\1& x_2 & x_2^2 &x_2^3 \\ 0 &1& 2x_1 & 3x_1^2 \end{bmatrix} $
is non-zero (for the system to be uniquely determined). I have calculated the value of the determinant using matlab, but the expression was pretty messy and it is not obvious to me that it must be non-zero (for different $x_0, x_1, x_2$).
Extra: How one can generalize that to a degree $n$ polynomial (i.e. to prove that given $n+1$ independent "conditions" there exists a unique polynomial that satisfies all of them).
Best Answer
Let $p_2$ be the unique polynomial of degree $2$ or less such that $f(x_i) = p_2(x_i)$ for $i = 0, 1, 2.$
Let $p_3(x) = (x - x_0)(x - x_1)(x - x_2),$ which is a polynomial of degree $3.$
Note that $p_3(x_1) = p_3(x_2) = p_3(x_3) = 0,$ and therefore for any constant real number $r,$ $f(x_i) = p_2(x_i) + rp_3(x_i)$ for $i = 0, 1, 2.$ Also note that the derivative of $p_2(x_i) + rp_3(x_i)$ is $p_2'(x_i) + rp_3'(x_i).$
Find $p_2'(x_1)$ and $p_3'(x_1).$ Let $p_2'(x_1) + r p_3'(x_1) = f'(x_1),$ where $r$ is a real constant. Solve for $r.$ Since $p_3'(x_1) \neq 0,$ the solution for $r$ is unique.
Set $p(x) = p_2(x) + r p_3(x),$ which is a polynomial of degree $3$ or less.