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Homework Help: Analysis Question 3

  1. Apr 28, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose f:(-1,1)→R is three times differentiable on the interval. Assume there is a positive value M so that ⎮f(x)⎮ ≤ M⎮x⎮³ for all x in (-1,1). Prove that f(0)=f'(0)=f"(0)=0.

    2. Relevant equations

    3. The attempt at a solution
    My professor started us off,
    ⎮f(0)⎮≤M(0)=0; f(0)=0
    f'(0)=lim as x→0 [(f(x)-f(0))/x-0 = lim as x→0 [f(x)/x].
    I know that ⎮f(x)/x⎮≤ 1/⎮x⎮(M⎮x⎮³
    ≤ Mx²
    Which means that f'(0) = 0
    I also know that the next step is to find f"(0) = lim as x→0 [(f'(x)-f'(0))/x-0].
    I need to know if f'(x) = 1/x?
  2. jcsd
  3. Apr 29, 2009 #2
    f'(x) is not 1/x. Think about what f(x) is for x > 0 and for x < 0. Then think about what f'(x) should be for x > 0 and for x < 0. This should help you continue the problem. When you have to show f''(0) = 0, again think about what f''(x) should be for x > 0 and for x < 0.
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