I won't be posting new problems for a couple of weeks. This one is nice:

Suppose $\displaystyle n \geq 1$ and $\displaystyle f: (a,b) \longrightarrow \mathbb{R}$ is a $\displaystyle C^n$ function (see here for the definition) and $\displaystyle f(x)f'(x)f''(x) \cdots f^{(n)}(x) = 0,$ for all $\displaystyle a < x < b.$ Show that $\displaystyle f$ is a polynomial of degree at most $\displaystyle n-1.$