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cinar

  • 2 years ago

Let f be a continuous and bounded function on [a,b] such that if

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  1. cinar
    • 2 years ago
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  2. cinar
    • 2 years ago
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    any idea?

  3. satellite73
    • 2 years ago
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    not really,but it kind of looks like a set up for integrating by parts

  4. cinar
    • 2 years ago
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    I guess so..

  5. cinar
    • 2 years ago
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    F'(x)=f(x) so we can say F'(t)=f(t) right..

  6. satellite73
    • 2 years ago
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    that is what i was thinking, yes

  7. satellite73
    • 2 years ago
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    maybe not exactly what you want, but you would get something like \(gF-\int f g'\)

  8. cinar
    • 2 years ago
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    \[\lim_{b \rightarrow} \int\limits_{0}^{b }G(t)F'(t)dt=\]

  9. cinar
    • 2 years ago
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    I guess enough to show that \[\int\limits f g'\] is convergent

  10. satellite73
    • 2 years ago
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    first part is no problem, since \(\lim_{t\to \infty}g(t)=0\) and \(\int_a^\infty f(t)\) is bounded

  11. satellite73
    • 2 years ago
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    yeah you hit the nail on the head, and frankly i don't see why that is true, so this might be the wrong approach

  12. satellite73
    • 2 years ago
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    because all you know about \(g'\) is that \(g'\leq 0\)

  13. satellite73
    • 2 years ago
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    does this come in a section after some theorem or lemma that maybe you are supposed to use?

  14. cinar
    • 2 years ago
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    no, this is homework, but prof gave the hint and said that you should use integration by part..

  15. satellite73
    • 2 years ago
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    well then i guess this is the right approach!

  16. cinar
    • 2 years ago
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