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Math Help - function behavior

  1. #1
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    Post function behavior

    Which of the following describes the behavior of the graph
    y=\sqrt[4]{|x-2|} at x=2?

    1. differentiable
    2. corner
    3. cusp
    4. vertical tangent
    5. discontinuity

    I think its a cusp..but not sure.
    Thanks!
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  2. #2
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    Quote Originally Posted by live_laugh_luv27 View Post
    Which of the following describes the behavior of the graph
    y=\sqrt[4]{|x-2|} at x=2?

    1. differentiable
    2. corner
    3. cusp
    4. vertical tangent
    5. discontinuity

    I think its a cusp..but not sure.
    Thanks!
    Cusp.

    Since the tangent does not exist at x = 2 but the curve is continuous, only options 2 and 3 are candidates. But if you look at the definition of each, the answer has to be cusp.
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  3. #3
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    Quote Originally Posted by live_laugh_luv27 View Post
    Which of the following describes the behavior of the graph
    y=\sqrt[4]{|x-2|} at x=2?

    1. differentiable
    2. corner
    3. cusp
    4. vertical tangent
    5. discontinuity

    I think its a cusp..but not sure.
    Thanks!
    The left and right derivatives go to \mp infinity as x goes to 2, which in the case of slopes are the same thing (the curve is vertical on both sides at x=1) which of course makes the point a cusp.

    CB
    Last edited by CaptainBlack; August 10th 2009 at 08:19 AM. Reason: fix typo
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post
    The left and right derivatives go to infinity as x goes to 2.

    CB
    Indeed. Left hand limit \rightarrow - \infty and right hand limit \rightarrow + \infty.
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