## Section 2.12 Exercises

###### 2

Find an example of a continuously differentiable function \(f:\mathbb R\to \mathbb R\) that attracts no critical point.
Hint

Draw a graph. Of course, you can't violate theorem TheoremĀ 2.6.2.

###### 3

Let \(f(x)=x^2-4x+5\text{.}\) Show that \(f\) has a super-attractive orbit of period 2.

###### 4

Let \(f(x)=3 x^2-6 x+3.415\text{.}\) Find all attractive orbits of \(f\text{.}\)

###### 5

Use software to find all orbits of period 2, 3, and 4 for \(f(x)=x^2-2\text{.}\)

###### 6

Use software to find one orbit of period 11 for \(f(x)=x^2-2\text{.}\)

###### 7

Let \(f_{\lambda}(x)=\lambda x(1-x)\text{.}\) Find the value of \(\lambda\) such that \(f_{\lambda}\) has a super-attractive orbit of period 3.

###### 8

Find an orbit of period 11 for the doubling map.

###### 9

Show that there is no orbit of the doubling map that is dense in some proper subinterval of \(H\) but not dense in \(H\) itself.