I'm trying to prove that a function $ f \in C^{\infty}(\mathbb{R}, \mathbb{R})$ that is strictly positive on $]-1,1[$ and equals $0$ everywhere else verifies the following statement : there exists $ m \in \mathbb{N}^*$ such that $f^{(m)} $ has strictly more than $m$ zeros in $]-1,1[$.
I tried some things that doesn't give me much information, for instance I know that there is a decreasing sequence $(x_p)_{p\geq2} \in ]-1,1[^{\mathbb{N}}$ so that $x_p \longrightarrow_{p \rightarrow \infty} x \in ]-1,1[$ and $\forall p \geq 2, f^{(p)}(x_p) = 0$.
Thanks for help.