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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?

1. $f^{(n)}$ has at least $10$ zeros, for $1 \leq n \leq 8$
2. $f^{(n)}$ has at least one zero, for $1 \leq n \leq 9$
3. $f^{(n)}$ has at least $10$ zeros, for $n \geq 10$
4. $f^{(n)}$ has at least one zero, for $n \geq 9$
asked in Calculus | 68 views

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