Derivatives of flat functions

Abstract

We remark that there is no smooth function $f(x)$ on $[0, 1]$ which is flat at $0$ such that the derivative $f^{(n)}$ of any order $n\geq 0$ is positive on $(0,1]$. Moreover, the number of zeros of the $n$-th derivative $f^{(n)}$ grows to the infinity and the zeros accumulate to $0$ when $n \to \infty$.