I think you're a little on the wrong track here. A fixed point is one that doesn't change after a transformation. Consider this simple example f(x)=x and the transformation f --> f^2. What points are fixed? In other words what values of x satisfy f(x)=f^2(x). The solution is fairly simple:
x=0 or x=1.
Now let's look at your problem:
The transformation is f --> -f, so:
Solve that and you'll have your answer.