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Definitely Maybe: Concavity and Point of Inflection

  1. Dec 2, 2009 #1
    1. The problem statement, all variables and given/known data
    f is a continuous function on [0, 8] and satisfies the following:
    Second Derivative Sign Test
    x ; f''
    0 [tex]\leq[/tex] x < 3 ; -
    3 ; 0
    3 < x < 5 ; +
    5 ; Does Not Exist
    5 < x < 6 ; -
    6 ; 0
    6 < x [tex]\leq[/tex] 8 ; -

    Based on this information is there a point of inflection at x = 5?
    (a) Definitely
    (b) Possibly
    (c) Definitely not

    2. Relevant equations
    Definition: A point of inflection is where the concavity changes and there is a tangent line.

    Concavity changes when f'' changes signs. Negative f'' means concave down; positive f'' means concave up.

    3. The attempt at a solution
    There is a sign change at f''(5), but the correct answer could either be "definitely" or "possibly." The point of inflection does not exist, but there is still a sign change, meaning it is a point of inflection. But my class and I were wondering whether it was possible that because there is no derivative at x = 5, there could also possibly be or not be a point of inflection there, too. The book says it's "possibly" a point of inflection, but going by the definition, we just presumed it to "definitely" be one.
     
  2. jcsd
  3. Dec 3, 2009 #2
    Can you think how this would reflect on the question? Can you think of two examples to satisfy the conditions on f''(x), one that does not have a tangent at x=5 and one which does?
     
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