Find set of polynoms

Question

How can I find all polynoms $P$ with degree $ \ge 3$ where

1. $P(2) = 1$
2. $P'(2) = 3$
3. $P''(2) = 4$
4. $P^{(n)}(2) = 0 \quad \forall n\geq 3$

Answer 1

Using the Taylor formula at $x=2$ we have that such a polynomial must be

\begin{align}P(x)&\\=&P(2)+P'(2)(x-2)+\dfrac{P''(2)}{2!}(x-2)^2+\dfrac{P'''(2)}{3!}(x-2)^3+\cdots+\dfrac{P^{(n)}(2)}{n!}(x-2)^{n}\\\underbrace{=}_{P^{(n)}(2)=0,n\ge 3}&P(2)+P'(2)(x-2)+\dfrac{P''(2)}{2!}(x-2)^2\\=&1+3(x-2)+2(x-2)^2.\end{align}

Answer 2

Hint: The derivative of $f(x)=c$ and $f(x)=0$ is zero, which means that the second derivative of your function is a constant function.