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Determine the exponent(s) α>0 so f'(0) exists

  1. Oct 1, 2011 #1
    Determine α>0 so that f'(0) exists



    [tex]f_{\alpha }(x)=|x|^{\alpha }sin\left (\frac{1}{x} \right ) , \left [x\neq 0, f_{\alpha }(x)=0 \right ][/tex]




    I derived the function in two cases, one where x<0 and one x>0, and saw that we get x in a denominator three times, As I understand it, it does not matter what α is since f'(0), atm, is undefined. I tried to move around and see if I could get rid of the x`s, but I failed.
    I also tried to see if I could substitute something(x=e^lnx), but again no luck.

    I do not think it is possible to try to find an inverse in some way, since in the end I think that I would get stuck with some x in the denominator.

    However, I'm still not sure how some α could fix the x`s in the denominators...
    All in all I´m quite lost and have no idea how to find α or(if thats the case) show that α is irrelevant
     
  2. jcsd
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