Suppose $f$ is continuous on $[0, \infty)$, differentiable on $(0, \infty)$ and $f(0) \geq 0$. Suppose $f'(x) \geq f(x)$ for all $x \in (0, \infty)$. Show that $f(x) \geq 0$ for all $x \in (0, \infty)$.
Suppose $f$ is continuous on $[0, \infty)$, differentiable on $(0, \infty)$ and $f(0) \geq 0$. Suppose $f'(x) \geq f(x)$ for all $x \in (0, \infty)$. Show that $f(x) \geq 0$ for all $x \in (0, \infty)$.