Is there a cubic polynomial $f(x)$ with real coefficients such that $f$ is monotonic and $f(x)=f^{-1}(x)$ has more than $3$ real roots?
Yep! Consider $f(x)=-(x+1)^3$. There are five real roots to $f(x)=f^{-1}(x)$.
Yep! Consider $f(x)=-(x+1)^3$. There are five real roots to $f(x)=f^{-1}(x)$.