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Cubic curve

  1. Feb 14, 2010 #1
    1. The problem statement, all variables and given/known data
    2) Given a cubic equation y = (x+5)(ax^2 + bx - 2). Give conditions on a and b for the equation to represent the following curve. The curve is attached to the email.
    http://img35.imageshack.us/img35/8246/question2o.png [Broken]

    2. Relevant equations



    3. The attempt at a solution
    I know that a>0.
    Since there are 3 intersections with the x-axis, then for (ax^2 + bx - 2), b^2 - 4ac > 0.
    b^2 - 4(a)(-2) > 0
    b^2 + 8a > 0
    b^2 > -8a

    If i differentiate it, I get
    3ax^2 + 2bx + 10ax + 5b - 2
    Since there are 2 stationary points,
    B^2 - 4ac = (2b + 10a)^2 - 4(3a)(5b-2) > 0
    b^2 - 5ab + 25a^2 + 6a > 0
    and I'm stucked again.
    Which approach is correct?

    Thanks.
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Feb 14, 2010 #2

    Mark44

    Staff: Mentor

    It seems to me that you are just about finished here. As you already said, a > 0, which you can tell from the behavior of the graph for very negative or very positive x.

    The inequality b2 > -8a is true for all real b, as long as a > 0. You were finding conditions on the discriminant so that there would be two real, distinct roots. If the discriminant had been equal to zero, there would have been a repeated root.
     
    Last edited by a moderator: May 4, 2017
  4. Feb 14, 2010 #3
    eh.. Why did I never thought of that?? Thank you! Haha.
     
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