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Math Help - Another Advanced Calculus Question (analysis)

  1. #1
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    Another Advanced Calculus Question (analysis)

    Let f(x) be continuous on [a,b]. Show that there exists a c in [a,b] such that f(c)=1/(b-a) times INTEGRAL(f(x)dx) (integral taken from a to b)
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    Proof:

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    My first thought is to use the Mean Value Theorem. BUT, the Mean Value Theorem requires that f be continuous AND differentiable. We are given continuity but not differentiability.

    Any IDEAS?
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  2. #2
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    Re: Another Advanced Calculus Question (analysis)

    Quote Originally Posted by computerproof View Post
    Let f(x) be continuous on [a,b]. Show that there exists a c in [a,b] such that f(c)=1/(b-a) times INTEGRAL(f(x)dx) (integral taken from a to b)
    Define $F(x) = \int_a^x {f(t)dt}$. Is $F$ continuous on $[a,b]$?

    Now use the mean value theorem.
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    Re: Another Advanced Calculus Question (analysis)

    F(x) must be continuous for other wise F '(x) = f(x) could not exist....is that correct?
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    Re: Another Advanced Calculus Question (analysis)

    Quote Originally Posted by computerproof View Post
    F(x) must be continuous for other wise F '(x) = f(x) could not exist....is that correct?
    Yes, $F'$ does exist and equals $f$.

    Now apply the MVT to $F$ on $[a,b]$.
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  5. #5
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    Re: Another Advanced Calculus Question (analysis)

    Thanks.....I really appreciate it
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