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Math Help - function inside function proofe question

  1. #1
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    function inside function proofe question

    it is given that f(x) is monotonically increasing and f(g(x)) is monotonically increasing

    does g(x) is monotonically increasing??

    i tried to solve it this way:

    i think that it does.so i want to disprove the theory that g(x) is not is monotonically increasing:

    suppose that g(x) is monotonically decreasing
    if a<b then g(a)>g(b)
    f(a)<f(b)
    f(g(a))<f(g(b))

    now what??
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  2. #2
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    Assuming that the derivatives exist.
    We know that f'(x)>0~\&~[f(g(x))]'=f'(g(x))g'(x)>0
    so what does that imply about g'(x)?
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  3. #3
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    that its positive too
    thanks
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