Let $I = (-a, a)$ be an open interval, for some $a > 0$, and let $K, f \in C^2(I)$. Assume that $K \leq 0$ and $f > 0$ on $I$. If $u \in C^2(I)$ satisfies the differential inequality
$$\begin{cases} u'(t) \geq K(t) + f(t)u(t), \quad t \in I \\ u(0) = 0 \end{cases},$$
can we say something about the signs of $u$ when $t < 0$ and when $t > 0$? What if $K \equiv 0$? I thought that maybe Gronwall's inequality may be useful, but how do I proceed?