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Math Help - derivative/real analysis

  1. #1
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    Oct 2011
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    derivative/real analysis

    Problem 1:

    Let f(x) be a real valued function defined for all x >= 1, that satisfies
    f(1) = 1 and
    f'(x) = 1/[x^2 + (f(x))^2]
    Prove that lim f(x) as x tends to infinity exists and is less than 1+ pi/4.

    Problem 2:

    Suppose that a continuously differentiable function f : IR --IR satisfies
    f'(x) = g(f(x)) + h(x) for x in IR,
    where the functions g, h : IR ---IR are C 00 (i.e. infinitely differentiable). Prove that
    the function f is infinitely differentiable as well.
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  2. #2
    Super Member girdav's Avatar
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    Rouen, France
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    Re: derivative/real analysis

    Problem 1
    f is increasing, and f'(x)\leq \frac 1{1+x^2}. Write f(x)-f(1)=\int_1^xf'(t)dt to get the result.

    Problem 2
    You have to use (or show) the fact that the composition of two smooth functions is smooth, and the sum of two smooth functions is smooth.
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