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Math Help - mvt and average values

  1. #1
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    mvt and average values

    exlain how you know that, if f is continuous on [a,b], then there is a c in (a,b) such that f(c) = f avg, the average value of f on [a,b]

    I have been able to show this by rearranging the mean value theorem for integrals

    but the question also offers a hint:
    Apply the mean-value theorem to g(x) = integral from a to x f(t) dt

    can someone show me the solution using this hint explicitly
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  2. #2
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    Quote Originally Posted by razorfever View Post
    exlain how you know that, if f is continuous on [a,b], then there is a c in (a,b) such that f(c) = f avg, the average value of f on [a,b]

    I have been able to show this by rearranging the mean value theorem for integrals

    but the question also offers a hint:
    Apply the mean-value theorem to g(x) = integral from a to x f(t) dt

    can someone show me the solution using this hint explicitly
    g(x) = \int_a^x f(t) dt

    g(b) = \int_a^b f(t) dt

    g(a) = 0

    by the MVT, there exists a c in the open interval (a,b) such that ...

    g'(c) = \frac{g(b) - g(a)}{b-a}

    g'(c) = \frac{\int_a^b f(t) dt - 0}{b-a}

    since g'(x) = f(x) by the FTC ...

    g'(c) = f(c) = \frac{\int_a^b f(t) dt}{b-a}
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