2018 Putnam A5

Let $f: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1)= 1$,

and $f(x) \geq 0$ for all $x \in \mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$

such that $f^{(n)}(x) < 0$.

令 $f: \mathbb{R} \to \mathbb{R}$ 为无限可微函数，满足 $f(0) = 0$, $f(1)= 1$,

和 $f(x) \geq 0$ 对于所有 $x \in \mathbb{R}$。证明存在一个正整数$n$和一个实数$x$

使得 $f^{(n)}(x) < 0$。