Show that there is $c \in (-1,2)$ such that $f'(c)=0$.

Question

$f:\mathbb{R} \rightarrow \mathbb{R}$ differentible function and $F:\mathbb{R} \rightarrow \mathbb{R}$ a primitive of $f$. We know $5F(xy^{2}-y)+F(x^{2}y-x)-F(x^{3}-y)F(2xy-2)\geq 9$ for every $x,y \in \mathbb{R}$. Show that there is $c \in (-1,2)$ such that $f'(c)=0$. I think that inequality somehow is related to a function that is positive and somehow we need to use Rolle's Thoerem with that function.