Results 1 to 2 of 2

Math Help - Is this function differentiable at zero?

  1. #1
    Junior Member
    Joined
    Jan 2010
    Posts
    39

    Is this function differentiable at zero?

    Is this function differentiable at zero?

    f(x)=\mid x\mid ^\frac{5}{2}

    I'm having a bit of trouble deciding whether functions like this are differentiable so steps would be helpful.
    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by featherbox View Post
    Is this function differentiable at zero?

    f(x)=\mid x\mid ^\frac{5}{2}

    I'm having a bit of trouble deciding whether functions like this are differentiable so steps would be helpful.
    Thanks.
    Since the value at 0 is 0, this function is differentiable at 0 if and only if \frac{f(x)}{x} has a limit as x\to 0. In the case of f(x)=|x|^\alpha, the ratio equals |x|^{\alpha-1} if x>0 and -|x|^{\alpha-1} if x<0 (because then |x|=-x), hence
    - it converges to 0 if \alpha>1;
    - if \alpha=1, it converges to 1 from the right and -1 from the left, hence it doesn't converge
    - if 0<\alpha<1, it diverges to +\infty from the right and -\infty from the left, hence it doesn't converge.

    As a conclusion, it is differentiable iff \alpha>1, and the derivative is 0 then.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Is the following function differentiable?
    Posted in the Calculus Forum
    Replies: 6
    Last Post: December 14th 2011, 10:38 AM
  2. Replies: 0
    Last Post: October 3rd 2010, 07:03 AM
  3. Differentiable function on Rn
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 14th 2010, 04:56 AM
  4. Differentiable function
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 13th 2009, 05:47 AM
  5. differentiable function
    Posted in the Calculus Forum
    Replies: 4
    Last Post: January 15th 2008, 04:30 AM

Search Tags


/mathhelpforum @mathhelpforum