Differential Inequality Help I have the inequality
$f''(x)x + f'(x) \leq 0$
Also, $f''(x)<0$ and $f'(x)>0$ and $x \in R^+$. And I need to figure out when it is true. I know it is a fairly general question, but I couldn't find any information in several textbooks I have skimmed. Also, I am not sure if integrating would require a sign reversal or not, so I cant go ahead and try to manipulate it my self.
Any help or mention of a helpful source would be much appreciated.
edit: forgot to mention $f(x)\geq 0$ for every $x \in R^+$

$$0\geq f''(x)x+f'(x)=(f'(x)x)'$$
that is why the function $$f'(x)x$$ decreases for positive reals. Then the maximum should be at $0$, so $$f'(x)x\leq 0,\quad x \in R^+$$ which is a contradiction.
