1. Help on Hint

hi I need some help understanding the hint this exercise gives:

Where $I=[-2,2]$ and
$A_1=\{x \in I : Q_c(x)\in I \}$.

$p_+$ is the fixed point of $Q_c$ in $[0,2]$.

$p_+=\frac{1+ \sqrt{1-4c}}{2}$

I dont understand why it suffices to find the c-values the hint points to guarantee the conclusion for $x \in [0,\frac{1}{2}]$

thank you!

2. Originally Posted by mabruka
hi I need some help understanding the hint this exercise gives:

Where $I=[-2,2]$ and
$A_1=\{x \in I : Q_c(x)\in I \}$.

$p_+$ is the fixed point of $Q_c$ in $[0,2]$.

$p_+=\frac{1+ \sqrt{1-4c}}{2}$

I dont understand why it suffices to find the c-values the hint points to guarantee the conclusion for $x \in [0,\frac{1}{2}]$

thank you!

Something seems to be missing in the question

Tonio

3. hmm ok.

We need to prove that $|Q^{\prime}_c(x)| > 1$ for all $x \ in [-2,2]$.

It is easy to see that it is true for $x > \frac{1}{2}$

To prove it for $x \leq \frac{1}{2}$ it is suggested to investigate which values of c
satisfy:

$Q_c(\frac{1}{2})< -p_+$

My question is:
Why it suffices to find the values of c which satisfy $Q_c(\frac{1}{2})< -p_+$ to show that $|Q^{\prime}_c(x)| > 1$ for $x \leq \frac{1}{2}$ ?

In other words, why does the hint "works" ?