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Math Help - check differentiability

  1. #1
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    Exclamation check differentiability

    need help...is the func. |x|^3 diff at x=0 or not?
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  2. #2
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    Quote Originally Posted by adhyeta View Post
    need help...is the func. |x|^3 diff at x=0 or not?
    Yes. f(x) = x^3 when x > 0 and f(x) = -x^3 when x \leq 0. Therefore f'(x) is continuus at x = 0.
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    MHF Contributor chisigma's Avatar
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    If you compute the derivative of f(x) in x=x_{0}as...

    \lim_{\delta \rightarrow 0} \frac{f(x_{0} + \delta)-f(x_{0})}{\delta}

    ... for f(x)= |x|^{3} and x_{0}=0 is...

    \lim_{\delta \rightarrow 0} \frac{|\delta|^{3}}{\delta} = \lim_{\delta \rightarrow 0} \delta\cdot |\delta|=0

    ... no matter which is the sign of \delta. So |x|^{3} is differentiable in x=x_{0}...

    Kind regards

    \chi \sigma
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