Let $I=[0,1]$ and consider continuous maps $F: I \rightarrow I$. Give an informal outline description of the two different bifurcations of fixed points of $F$ that can occur.

Illustrate your discussion by considering in detail the logistic map

$$F(x)=\mu x(1-x),$$

for $\mu \in(0,1+\sqrt{6}]$.

Describe qualitatively what happens for $\mu \in(1+\sqrt{6}, 4]$.

[You may assume without proof that

$$x-F^{2}(x)=x(\mu x-\mu+1)\left(\mu^{2} x^{2}-\mu(\mu+1) x+\mu+1\right)$$