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Math Help - Proving a statement involving the intermediate value theorem

  1. #1
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    Proving a statement involving the intermediate value theorem

    The question:

    Suppose that f is continuous on [0,1] and that Range(f) is a subset of [0,1]. By using g(x) = f(x) - x, prove that there is a real number c in [0,1] such that f(c) = c.

    I'm not sure where to start. I know that the intermediate value theorem applies, since we're given a function g(x) which is continuous on the interval [0,1]. However, I don't know how to use this to prove that f(c) = c.

    Any help would be great.
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  2. #2
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    What can be said about the values of g(0)~\&~g(1)?.

    Recall that you are given 0\le f(x) \le 1.
    Last edited by Plato; August 7th 2010 at 04:23 AM. Reason: correct typo
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  3. #3
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    I'm not sure what can be said. I don't know what f(x) is, I just know part of its domain and range. I'm also not sure how we are given 0\le f(x) \le 0
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  4. #4
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    That was a typo. Plato meant to say 0\le f(x)\le 1. (That may be the first mistake he has ever made!)

    First, if either f(0)=0 or f(1)= 1, we are done. So we can assume that f(0)> 0 and that f(1)< 1.

    Let g(x)= f(x)- x as you say. Then g(0)= f(0). Is that positive or negative?

    g(1)= f(1)- 1. Is that positive or negative?
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  5. #5
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    g(0) would be positive, g(1) would be negative.
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  6. #6
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    Quote Originally Posted by Glitch View Post
    g(0) would be positive, g(1) would be negative.
    Correct. Now what?
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  7. #7
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    Hmm, I'm not sure. I feel like I'm missing something obvious here...
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  8. #8
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    g(0)>0~\&~g(1)<0.

    Because g is continuous apply the MVT.
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