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Math Help - Curve Analysis (y=x-3x^1/3)

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    Curve Analysis (y=x-3x^1/3)

    My teacher has given some test prep/homework questions. She says fully analyze them which means find the intercepts, symmetry, asymptotes, intervals of inc/dec, local max and min, inflection points, and concavity. Here are a couple of questions from the sheet. The first one I've gotten part way through (I put up my rough copy of it) and the second one I can't get because of the double root.



    Here's my rough copy of the attempt at solution for the first one but I know that parts of it are wrong:
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    Re:

    RE:

    #17.

    Attached Thumbnails Attached Thumbnails Curve Analysis (y=x-3x^1/3)-concavity.gif  
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    #21 Part #1:

    RE: #21 Part #1:

    Attached Thumbnails Attached Thumbnails Curve Analysis (y=x-3x^1/3)-increase-decrease-part-1.jpg  
    Last edited by qbkr21; April 28th 2007 at 10:15 PM.
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    #21 Part #2:

    RE:#21 Part #2:
    Attached Thumbnails Attached Thumbnails Curve Analysis (y=x-3x^1/3)-increase-decrease-part-2.jpg  
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    I got a couple of questios. For the first one (#17):
    Is there no point of inflection?
    And when I plugged in -1, I get f(-x)= f(x) therefore is the function not symmetric about the x-axis instead of origin?

    For the 2nd one (#21), I see that there isn't a point of inflection but is it symmetric about the origin or x-axis?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by SportfreundeKeaneKent View Post
    I got a couple of questios. For the first one (#17):
    Is there no point of inflection?
    And when I plugged in -1, I get f(-x)= f(x) therefore is the function not symmetric about the x-axis instead of origin?

    For the 2nd one (#21), I see that there isn't a point of inflection but is it symmetric about the origin or x-axis?
    Here's the answer to the question you had on #17
    Attached Thumbnails Attached Thumbnails Curve Analysis (y=x-3x^1/3)-17.gif  
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by SportfreundeKeaneKent View Post
    And when I plugged in -1, I get f(-x)= f(x) therefore is the function not symmetric about the x-axis instead of origin?
    f(x) = x - 3x^(1/3)

    now f(-x) = (-x) - 3(-x)^(1/3) = -x + 3x^(1/3) = -f(x)
    since f(-x) = -f(x), we are symmetric about the x-axis and not symmetric about the y-axis and origin

    see Reflections and Symmetry it tells you the different tests for symmetry, you can check them yourself. there is a pink table in the middle of the page entitled "Types of symmetry" under the sub-heading "Ways to test different symmetry" in big bold blue letters, use this table to answer your questions about 21, if you have problems, say so
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