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Math Help - Generalized Mean Value Theorem

  1. #1
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    Generalized Mean Value Theorem

    Let a<b be elements of R, and let f,g:[a,b]-->R be continuous on [a,b] and differentiable on ]a,b[. Assume that f(a) is less than or equal to g(a) and that f'(x)<g'(x) for x element of ]a,b[. Show that f(x)<g(x) for x element of ]a,b].

    Can this be done using Generalized Mean Value Theorem? And either way, could somebody give me some pointers?

    Thanks a bunch!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by TexasGirl
    Let a<b be elements of R, and let f,g:[a,b]-->R be continuous on [a,b] and differentiable on ]a,b[. Assume that f(a) is less than or equal to g(a) and that f'(x)<g'(x) for x element of ]a,b[. Show that f(x)<g(x) for x element of ]a,b].

    Can this be done using Generalized Mean Value Theorem? And either way, could somebody give me some pointers?

    Thanks a bunch!
    Consider h:[a,b] \rightarrow \manthbb{R} such that

    \forall x \in [a,b]\ h(x)=f(x)-g(x).

    Then h is strictly decreasing in [a,b], and h(a)\le 0.

    And the rest should be trivial.

    RonL
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  3. #3
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    thanks

    thanks a bunch =)
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