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Inflection Points and Intervals

  1. Nov 3, 2013 #1

    Qube

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data

    Suppose that a continuous function f(x) has horizontal tangent lines at x = -1, x = 0, and x = 1. If f"(x) = 60x^3 - 30x, then which of the following statements is/are true?

    A) f(x) has a local max at x = 1
    B) f(x) has a local min at x = -1
    C) f(x) has an inflection point at x = 0

    2. Relevant equations

    Local maximums occur at critical points.

    All points at which horizontal tangent lines occur are critical points because the existence of a horizontal tangent line at that point implies the existence of that point on the function, and as we know, critical points must exist in the domain of the function.

    Therefore, x = 1, 0, and 1 are critical points.

    We can use the second derivative test to test for local extrema.

    3. The attempt at a solution

    f"(-1) = - 30. x = -1 is a local max. B is true.

    f"(1) = 30. x = 1 is a local min. A is true.

    f(x) has an inflection point at x = 0; the second derivative is 0 at x = 0 and x = ±1/sqrt(2).

    f"(x) changes sign around x = 0 from being positive in the interval (-1/sqrt(2), 0) and (0, 1/sqrt(2)).

    Therefore, all three are true.
     
  2. jcsd
  3. Nov 3, 2013 #2

    Mark44

    Staff: Mentor

    Your reasoning looks good to me.
     
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