Let $f : \mathbb{R} \longrightarrow \mathbb{R}$ be a twice-differentiable function such that $f\left(\frac{1}{n}\right) = 0$ for every positive integer $n$. Choose the correct statement(s) from below:\\
(A) $f(0) = 0$;\\
(B) $f'(0) = 0$;\\
(C) $f''(0) = 0$;\\
(D) $f$ is a nonzero polynomial.