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Math Help - Question about the concavity of graphs

  1. #1
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    Question about the concavity of graphs

    Find where the graph is concave up and concave down.
    3X^4 - 16X^3 + 24x^2 + 1

    I determined that there were points of inflection at x=2/3 and x=2. At x=2/3 the graph is concave up. At x=2 , the graph is concave down.
    If I wanted to state where the graph is concave up or concave down, would it be correct to say that the graph is concave up on the interval (2/3, 2) and concave down on the interval (2, 00]
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  2. #2
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    Quote Originally Posted by Kenneth View Post
    Find where the graph is concave up and concave down.
    3X^4 - 16X^3 + 24x^2 + 1

    I determined that there were points of inflection at x=2/3 and x=2. At x=2/3 the graph is concave up. At x=2 , the graph is concave down.
    If I wanted to state where the graph is concave up or concave down, would it be correct to say that the graph is concave up on the interval (2/3, 2) and concave down on the interval (2, 00]
    Hi Kenneth!


    I worked the problem out for you please click on the following 2 links:

    1. http://item.slide.com/r/1/92/i/Gpz_9...C2_-tk731Iv0i/
    2. http://item.slide.com/r/1/92/i/76ODL...OhHtxGKu0sIch/

    Best of Luck!
    Last edited by qbkr21; March 8th 2007 at 05:07 PM.
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  3. #3
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    Quote Originally Posted by qbkr21 View Post
    Hi Kenneth!


    I worked the problem out for you please click on the following 2 links:

    1. http://item.slide.com/r/1/92/i/Gpz_9...C2_-tk731Iv0i/
    2. http://item.slide.com/r/1/92/i/76ODL...OhHtxGKu0sIch/

    Best of Luck!
    oh, the number line. I understand now.

    Thanks.
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  4. #4
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    Hello, Kenneth!

    Find where the graph is concave up and concave down:
    . . f(x) .= .3x^4 - 16x + 24x + 1

    I determined that there were points of inflection at x = 2/3 and x = 2 . . . yes!
    At x = 2/3, the graph is concave up . . . no
    At x = 2 , the graph is concave down . . . no
    At a point of inflection, f''(x) = 0.
    . . The curve is neither concave up nor concave down.
    An inflection point is where the concavity changes.



    If I wanted to state where the graph is concave up or concave down,
    would it be correct to say that the graph is concave up on the interval (2/3, 2)
    and concave down on the interval (2, ∞)> . . . no
    The two inflection points create three intervals to test.

    . . - - - - * - - - - - * - - - -
    . . - - - -2/3- - - - -2


    The second derivative is: .f''(x) .= .12(3x - 8x + 4)

    On (-∞, 2/3), try x = 0
    . . f''(0) .= .12(30 - 80 + 4) .= .+48 . concave up: U

    On (2/3, 2), try x = 1
    . . f''(1) .= .12(31 - 81 + 4) .= .-12 . concave down: ∩

    On (2, ∞), try x = 3
    . . f''(3) .= .12(33 - 83 + 4) .= .+84 . concave up: U


    Therefore: .concave up on (-∞, 2/3) U (2, ∞)
    . . . . . . . . concave down on (2/3, 2)

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