I am trying to solve this question, here is my solution, is it correct?
Question: Let $f(x)=-\frac{1}{2}x^2-x+\frac{1}{2}$. Show that $1$ is an attracting period-2 point.
Solution:
$$f(1)=-1$$$$f(-1)=f(f(1))=1$$
Hence $x=1$ is a period-2 point. Since
$$f^\prime(x)=-x-1$$
then
$$|f^{\prime}(1)f^{\prime}(-1)|=|(-2)\times 0|=0<1$$
Hence $x=1$ is an attracting period-2 point.