Functional inequality $f(x)≥f(x+yf(x))(y+1)$

Question

Show that if function $f$ satisfies $f(x)\geq f(x+yf(x))(y+1)$ for $x,y>0$ then $f(x)>0$ is false. It is clear that $f$ is non-increasing but I can't show that it will pass OX and not assuming $f$ is differentiable.