if $f(x)f(y)+f(xy)\le -\frac{1}{4},\forall x,y\in[0,1)$ show that $f(x)=-\frac{1}{2}$

Let function $ f:[0,1)\to R$ ,such $ $ f(x)f(y)+f(xy)\le -\dfrac{1}{4},\forall x,y\in[0,1)$ $ show that $ $ f(x)=-\dfrac{1}{2}$ $

I have solve $ f(0)=-\dfrac{1}{2}$ ,only let $ x=y=0$ ,we have $ $ f^2(0)+f(0)\le-\dfrac{1}{4}\Longrightarrow (f(0)+\frac{1}{2})^2\le 0\Longrightarrow f(0)=-\dfrac{1}{2}$ $ But I can’t prove $ f(x)$ be constant,Thanks