Consider the logistic map $F(x)=\mu x(1-x)$ for $0 \leqslant x \leqslant 1,0 \leqslant \mu \leqslant 4$. Show that there is a period-doubling bifurcation of the nontrivial fixed point at $\mu=3$. Show further that the bifurcating 2 -cycle $\left(x_{1}, x_{2}\right)$ is given by the roots of

$$\mu^{2} x^{2}-\mu(\mu+1) x+\mu+1=0 .$$

Show that there is a second period-doubling bifurcation at $\mu=1+\sqrt{6}$.