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Math Help - Fixed Point Problem

  1. #1
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    Fixed Point Problem

    Define f:[1, \infty ] \rightarrow R by f(x) = 1+ \sqrt{x} for all x \geq 1. Show that f has exactly one fixed point.
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  2. #2
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    You do know that f is strictly increasing on [1,\infty )?
    The find a solution to 1 + \sqrt t  = t where t>1.
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  3. #3
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    Okay, the solution I found was t=1, but you said t > 1. I can't find a solutoin that matches this condition.
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    Quote Originally Posted by tttcomrader View Post
    Okay, the solution I found was t=1.
    How in the world is t=1 a solution?
    1 + \sqrt 1  = 2 \ne 1.
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  5. #5
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    Quote Originally Posted by tttcomrader View Post
    Define f:[1, \infty ] \rightarrow R by f(x) = 1+ \sqrt{x} for all x \geq 1. Show that f has exactly one fixed point.
    If x \in [0, \infty) is a fixed point of f then:

    <br />
x=f(x)=1+\sqrt{x}<br />

    Now introduce y=\sqrt{x}, solve the equation you get for y, and then sort the roots according to if they correspond to valid solutions of the original fixed point problem.

    RonL
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