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Math Help - derivatives of functions

  1. #1
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    derivatives of functions

    I appreciate if you can help me guys with any of these problems. They are good for practice and review of derivatives.

    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.

    problem 3:


    Prove that if f : [0,1)--- IR is nonnegative, integrable, and uniformly
    continuous, then lim f(x) =0 as x tends to 00.

    Problem 4:

    Suppose that a differentiable function f : IR ---IR and its derivative f'
    have no common zeros. Prove that f has only finitely many zeros in [0, 1].

    Problem 5:


    Suppose that f : [0,1)---IR is continuous on [0,1), differentiable on
    (0,00), f(0) = 0, and lim f(x) = 0 as x tends to 00. Prove that there exists a point c in (0,00) such
    that f'(c) = 0.
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  2. #2
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    Re: derivatives of functions

    Quote Originally Posted by skybluesea2010 View Post
    I appreciate if you can help me guys with any of these problems. They are good for practice and review of derivatives.

    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.

    problem 3:

    Prove that if f : [0,1)--- IR is nonnegative, integrable, and uniformly
    continuous, then lim f(x) =0 as x tends to 00.

    Problem 4:

    Suppose that a differentiable function f : IR ---IR and its derivative f'
    have no common zeros. Prove that f has only finitely many zeros in [0, 1].

    Problem 5:

    Suppose that f : [0,1)---IR is continuous on [0,1), differentiable on
    (0,00), f(0) = 0, and lim f(x) = 0 as x tends to 00. Prove that there exists a point c in (0,00) such
    that f'(c) = 0.
    Please don't post more than two questions in a thread. Otherwise the thread can get convoluted and difficult to follow. See rule #8: http://www.mathhelpforum.com/math-he...hp?do=vsarules.

    Thread closed.
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