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Analysis Homework Help

  1. Nov 30, 2006 #1
    1. The problem statement, all variables and given/known data
    Suppose f and g are differentiable on R, and f(a) = g(a) and f'(x) <= g'(x) for all x >= a. Show that f(x) <= g(x) for all x >= a. Give a physical interpretation of this result.

    Also, using the Mean Value Theorem:
    (a) Let f: R --> R be a differentiable function. Suppose that its derivative f'(x) is bounded Prove that f is uniformly continuous.
    (b) Let f: R --> R be a differentiable function. Suppose that lim (x --> infinity) f'(x) = infinity. Show that f cannot be uniformly continuous.
    (c) Let g(x) = (x)^1/2 show that g'(x) is unbounded on (0,1] but g(x) is uniformly continuous on [0,1].

    2. Relevant equations



    3. The attempt at a solution
    I don't know how to quite formulate the inequalities, any help?
     
  2. jcsd
  3. Nov 30, 2006 #2

    AKG

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    Consider the function f-g.
     
  4. Dec 1, 2006 #3
    i still dont see it, any more hints? thanks
     
  5. Dec 1, 2006 #4

    AKG

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    (f-g)(a) = 0, (f-g)'(x) < 0 for all x > a. Use MVT to prove that (f-g)(x) < 0 for all x > a.
     
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