finding a function with predetermined $f^{(n)}$s at $0$ and $1$. Let $a_n,b_n$ be 2 arbitrary sequences of real numbers. Is there any $C^\infty$ function $f:\mathbb{R}\to\mathbb{R}$ such that for any $n$: $$f^{(n)}(0)=a_n,\quad f^{(n)}(1)=b_n$$

For smooth functions, this is a version of Whitney's extension theorem.

For analytic functions, the answer is 'no' as already stated in the comments. It's not even true for one point.