Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function that vanishes at $10$ distinct points in $\mathbb{R}$. Suppose $f^{(n)}$ denotes the $n$-th derivative of $f$, for $n \geq 1$. Which of the following statements is always true?
- $f^{(n)}$ has at least $10$ zeros, for $1 \leq n \leq 8$
- $f^{(n)}$ has at least one zero, for $1 \leq n \leq 9$
- $f^{(n)}$ has at least $10$ zeros, for $n \geq 10$
- $f^{(n)}$ has at least one zero, for $n \geq 9$