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Use Taylor's theorem to show a function is differentiable at x=0

  1. May 27, 2012 #1
    1. The problem statement, all variables and given/known data

    Use Taylor's theorem to estimate |(ex)-x-1| for 0≤x≤1. Thus prove that if a>(1/2) then:

    f(x)=(1-|x|a)*(ex)a is differentiable at x=0

    2. Relevant equations



    3. The attempt at a solution

    So |(ex)-x-1|=(x^2)/2+(x^3)/6+(x^4)/24...

    But I don't see how this helps, I have considered using the Lagrange remainder as well but again I can't see how that would help either. Any help would be greatly appreciated.
     
  2. jcsd
  3. May 27, 2012 #2

    HallsofIvy

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    Staff Emeritus
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    Since the lowest power of x in that formula is 2 so, for x less than 1, it is less than x.
     
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