## 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