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Math Help - monotone increasing

  1. #1
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    Question monotone increasing

    Let f : [a, b] -> R be a positive continuous function, i.e., f(x) >= 0 for all x in [a, b].

    I want to show that the function F : [a, b] -> R defined as F(x) = integral of f with limits x and a, for x in [a, b] is monotone increasing (i.e., F(x) >= F(y) if x >= y).

    any ideas? thankz
    Last edited by dopi; November 5th 2006 at 10:34 AM. Reason: error in title
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  2. #2
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    Quote Originally Posted by dopi View Post
    Let f : [a, b] -> R be a positive continuous function, i.e., f(x) >= 0 for all x in [a, b].

    I want to show that the function F : [a, b] -> R defined as F(x) = integral of f with limits x and a, for x in [a, b] is monotone increasing (i.e., F(x) >= F(y) if x >= y).

    any ideas? thankz
    Okay, you have the function
    F(x)=\int_x^a f(t) dt
    It exists since all continous functions are integrable.
    Now, what ails me is that the upper limit is a, I think you wanted to write c\in [a,b] thus hence the function,
    F(x)=\int_x^c f(t)dt=-\int_c^x f(t)dt
    The derivative of this function is (2nd fun. the. of cal.)
    F'(x)=-f(x)<0
    Thus the function is decreasing NOT increasing.

    Thus, I think you wanted to write,
    F(x)=\int_c^x f(t)dt
    Then using the above concepts it is increasing on [a,b]
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  3. #3
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    Question verifying

    Quote Originally Posted by ThePerfectHacker View Post
    Okay, you have the function
    F(x)=\int_x^a f(t) dt
    It exists since all continous functions are integrable.
    Now, what ails me is that the upper limit is a, I think you wanted to write c\in [a,b] thus hence the function,
    F(x)=\int_x^c f(t)dt=-\int_c^x f(t)dt
    The derivative of this function is (2nd fun. the. of cal.)
    F'(x)=-f(x)<0
    Thus the function is decreasing NOT increasing.

    Thus, I think you wanted to write,
    F(x)=\int_c^x f(t)dt
    Then using the above concepts it is increasing on [a,b]
    With this question that i posted i denoted F(x) = integral of f with upper limit x and lower limt alpha, but you have added (t) as in f(t), where as iv just wrote f, so i think it might be an riemann integra....does this affect the ansower of the question to what you have done?
    thankz
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