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Math Help - Convergent Sequences and Continuity Proof

  1. #1
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    Convergent Sequences and Continuity Proof

    Don't really know how to prove this at all, I'm much appreciative of any help.

    Let f : [a,b] ⇒ ℝ be a continuous function. Show that there exists a w in [a,b] such that

    a
    ∫ f(t)dt = f(w)(b-a)
    b

    Here is the hint that i need to use to solve this problem:
    f attains a smallest value A and a largest value B. Show that:

    -----------a
    A ≤ 1/(b-a)∫ f(t)dt ≤ B
    -----------b

    Thank you again for the help!
    Last edited by ml692787; September 28th 2007 at 07:44 PM. Reason: mistype
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  2. #2
    MHF Contributor red_dog's Avatar
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    So we have \displaystyle A\leq\frac{1}{b-a}\int_a^bf(t)dt\leq B.
    Let \alpha=\frac{1}{b-a}\int_a^bf(t)dt. Then \alpha is a mean value of f.
    f is continuous on [a,b], then f has the Darboux property. So, exists w\in[a,b] such as f(w)=\alpha.
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