The following practice problem was given:
Find all polynomials $p(x)$ that satisfy the interpolation conditions:
- $p(0)=1$
- $p'(-1)=p(1)$
and the degree of $p(x)$ as small as possible.
I don't know how to tackle this problem. Because of the derivative I thought I could use Hermite interpolation, but since there is no concrete point for the derivative, I don't know how to proceed.
$$ a_0+a_1x_0+a_1x_0^2=f(0)\\ a_0+a_1x_1+a_1x_1^2=f(1)\\ a_1+2a_1x_2=f(1) $$
I constructed this system, but again, no idea if this is in the right direction.
If the polynomial $p$ is constant, say $p(x) = c$, then from $p(0) = 1$, we get $c=1$. However, the second condition is not fulfilled, since $p'(-1) = 0 \neq 1 =p(1)$.
Let us try with a polynomial of degree 1, say $p(x) = ax+b$. Then $p'(x) = a$. From $p(0) = 1$, we get that $b=1$, and from $p'(-1) = p(1)$, we get that $a = a+b$, i.e. $b=0$, which is a contradiction.
Hence, let us try with a polynomial of degree 2, say $p(x) = ax^2+bx+c$. Then $p'(x) = 2ax+b$. Thus, from $p(0) = 1$ we have $c=1$ and from $p'(-1) = p(1)$, we get that $-2a+b = a+b+c$ and hence $a=\frac{-1}{3}$. There is no condition for $b$.
Thus the polynomials we are looking for are $$p(x) = \frac{-1}{3}x^2 +bx+1$$ with $b \in \mathbb{R}$.