The question is in the title, really. More precisely, suppose $F:[0,1] \to \mathbb{R}$ is continuously differentiable and satisfies
(i) F(t)F'(t) \le 0 for all $t \in [0,1]$
(ii) $F(0) = 1$
Does this imply that $F(t) \geq 0$ for all $t \in [0,1]$?
My intuition is that (i) implies that $F$ and F' have different signs. Since $F(0) = 1$, this means that $F$ must cross the $t$-axis at some point, but the instant it becomes negative it would have to immediately shoot up again. I think this is not possible, but I have not been able to come up with a proof or counter-example. Thanks for your help!