On differentiable functions on real line satisfying $f'(x)\ge f(x)^2 , \forall x>0$

Question

Does there exist a real valued differentiable function $f$ on real line such that $f'(x) \ge f(x)^2 , \forall x >0$ ? If such a function exist , must it be twice differentiable or at least Lipschitz ? Please help . Thanks in advance

Answer 1

If you restrict to equality, you have $$ \frac{du}{dt} = u^2, $$ which separates to $$ u^{-2}du = dt $$ and integration yields $$ \frac{-1}{u} = t+C $$ so $$ u = \frac{-1}{t+C} $$ should do.