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Math Help - differentiability

  1. #1
    Senior Member Sambit's Avatar
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    Question differentiability

    Let f:R-->R be a function defined as
    f(x)=x^{3/2},x>=0
    -|x|^{3/2}, x<0.

    What can be said about its differentiability? Is is differentiable at 0? If yes, how many times?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Verify f'_{+}(0)=f'_{-}(0)=0 so , :

    f'(x)=\begin{Bmatrix}{ 3\sqrt{x}/2}&\mbox{ if }& x>0\\0 & \mbox{if}& x=0\\\ldots & \mbox{if}& x<0\end{matrix}

    Now, you can study the existence or not of f''(0) .


    Fernando Revilla
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  3. #3
    Senior Member Sambit's Avatar
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    I am getting f(x)=x^{3/2},x>=0
    -x^{3/2},x<0
    which is differentiable.

    Differentiating, f'(x)=\frac{3}{2}x^{1/2}, x>=0
    \frac{3}{2}(-x)^{1/2},x<0

    This is also continuous and differentiable at 0, and so on. So I think it is correct that the function is infinitely differentiable. Am I right?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Sambit View Post
    So I think it is correct that the function is infinitely differentiable. Am I right?

    No, prove that:

    \displaystyle\lim_{h \to 0^+}{\dfrac{f'(h)-f'(0)}{h}}=\ldots=+\infty

    so, f''(0) does not exist.


    Fernando Revilla
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  5. #5
    Senior Member Sambit's Avatar
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    Okay....got it. thank you.
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