If the k-th derivative of a function is bounded away from zero on an interval how many zeroes can the function have in the interval? If $f:[a,b] \to \mathbb{R}$ is $C^k$ and $f^{(k)}(x) \geq 1$ on $[a,b]$, how many zeroes can $f$ have in the interval $[a,b]$? Can it have infinitely many? 

By Rolle's Theorem, if we have a differentiable function $g:[a,b] \to \mathbb{R}$ with $g(x) = g(y)$, there exists $c \in [x,y]$ such that $g'(c) = 0$. In particular, if $f$ has infinitely many zeroes, so does $f'$, and by induction so does $f^{(k)}$, which contradicts the hypothesis. With this exact same argument you can give an upper bound on the zeroes of $f$. If $f^{(k)}$ has no zeroes, $f^{(k-1)}$ can't have more than one, and with the same argument $f^{(k-2)}$ can't have more than two, and so on. In the end, we get that $f$ has at most $k$ zeroes.
