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Math Help - Function: Find Increasing/Decreasing, Concavity up down. etc.

  1. #1
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    Function: Find Increasing/Decreasing, Concavity up down. etc.

    f(x)=x^3-8x+16/x

    For the following function, find the

    Domain: all reals but 0
    Symmetry: Origin Symmetry
    intercepts - y - none
    - x = +-2
    relative extrema :
    Taking f'(x) and solving for 0 to get the critical numbers
    f'(x) = 3x^2 - 16/x^2 - 8
    setting = to 0, gives critical numbers of +-2
    Plugging in critical numbers into f(x)
    f(-2) = 0
    f(2) = 0
    can't tell which is max or min, so make a number line, test points to left of -2, in between -2 and 2, and to the right of 2
    f(-3) = -8.333
    f(-2) = 0
    f(-1) = -9
    f(1) = 9
    f(2)=0
    f(3) = 8.333

    This means that a maximum is at -2,0 and minimum at 2,0

    Is that correct for the relative extrema?
    intervals of increasing/decreasing behavior
    For the intervals, once again, i found the critical number, made the number ilne and picked test points
    From (-∞ to -2) the graph was increasing, (-2 to 0) decreasing , (0 to 2) decreasing, and (2 to ∞) increasing

    whats the right way to do this problem
    inflection points
    i know we need to take the second derivative and set = to 0 but it is not working out? Are there no inflection points? and if there are no inflection points, how can there be any intervals of concavity? Can there?

    f"(x) = 6x+32/x^3
    cant solve for f"(x) = 0

    What do i do?
    intervals of concave up/down
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  2. #2
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    Quote Originally Posted by NandP View Post
    f(x)=x^3-8x+16/x
    Hi NandP,

    Is x=0 part of the function domain?

    What is f(0) ?

    This will explain your conundrum about inflexion points.
    What is the shape of the graph now, given that your work is correct.
    You can now deduce your concave regions.
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  3. #3
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    f(0) is undefined. What does that mean about the inflection points though? I dont think there are any inflection points.. and as for concavity, i think the graph just concaves down from -∞ to 0 and concaves up from 0 to ∞..if you graph it, its just two parabolas, one opening down from -∞ to 0 and one opening up from 0 to ∞,

    i sitll dont know what to do for the last two, inflection and intervals
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  4. #4
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    Sorry, I don't normally use terms like "undefined",
    since it doesn't really say much.

    For your function, x=0 causes 16/x to shoot to infinity,
    plus infinity for x>0 and minus infinity for x<0.
    f(x) approaches infinity because the 16/x term is dominant for x near zero.

    Your graph 'looks like' a pair of parabolas.
    Last edited by Archie Meade; January 8th 2010 at 01:10 AM.
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