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Math Help - need help

  1. #1
    Newbie
    Joined
    Jan 2010
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    5

    need help

    I need help in the following questions :

    Qn1 : Let f ; g be continuous functions from [0, 1] onto [0, 1]. Prove that there is a ∈ [0, 1] such that f(g(a)) = g(f (a)).

    Qn2 : Compute the limit lim(n→∞) integrate (0,pi,cos(x^n))

    Thanks in advance.
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  2. #2
    Senior Member Dinkydoe's Avatar
    Joined
    Dec 2009
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    411
    (1)

    First observe that g\circ f and f\circ g are onto since f and g are.

    Suppose that  g(f(x))>f(g(x)) for all x\in[0,1]. This means that g(f(x_0))> 1 for some x_0\in[0,1]. And g(f(x))< f(g(x)) for all x implies there exists a x_0\in [0,1] such that g(f(x_0)) < 0.

    These are contradictions, hence the conclusion follows. (By continuity)

    (You can use intermediate value theorem to show: f(g(x))-g(f(x)) < 0 and f(g(x))-g(f(x))> 0 for some x implies there exists a a\in [0,1] such that f(g(a))-g(f(a)) = 0, by continuity of f(g(x))-g(f(x))).
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