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Thread: Help on Hint

  1. #1
    Member mabruka's Avatar
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    Help on Hint

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




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

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

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


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

    thank you!
    Last edited by mabruka; Feb 21st 2010 at 06:12 PM.
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  2. #2
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    Quote Originally Posted by mabruka View Post
    hi I need some help understanding the hint this exercise gives:




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

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

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


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

    thank you!

    Something seems to be missing in the question

    Tonio
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  3. #3
    Member mabruka's Avatar
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    hmm ok.


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


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

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

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

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

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