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Queries regarding Inflection Points in Curve Sketching

  1. Nov 15, 2014 #1
    1. The problem statement, all variables and given/known data

    Let A be a set of critical points of the function f(x).
    Let B be a set of roots of the equation f''(x)=0.
    Let C be a set of points where f''(x) does not exist.
    It follows that B∪C=D is a set of potential inflection points of f(x).

    Q 1: Can there exist any inflection points of f(x) outside the set D?
    Q 2: If c is a point such that c∈D and c∉A, is it necessarily an inflection point?

    2. Relevant equations, theorems, and definitions

    - Critical points are defined as those points inside the domain of f(x) where f'(x)=0 or f'(x) does not exist.

    - Points of inflection are defined as the points in the domain of f(x) where f''(x) changes sign. At these points, the function changes concavity.

    - If c is a critical point of f(x), then the following are true:
    -- If f''(c) > 0, then the curve is concave up at c.
    -- If f''(c) < 0, then the curve is concave down at c.
    -- If f''(c) = 0 or f''(c) does not exist, then no conclusion can be made about concavity of the curve at c without further information.

    3. The attempt at a solution

    Q 1: Can there exist any inflection points of f(x) outside the set D?
    Ans: I couldn't find a satisfactory answer or implication for this query in the various calculus texts that I have been referring to. In my experience, such an inflection point is non-existent.

    Q 2: If c is a point such that c∈D and c∉A, is it necessarily an inflection point?
    Ans: Obtaining a point anywhere in the solution of a problem is usually suggestive of something significant happening at that point. While solving a curve-sketching problem, the points we obtain are normally classified as a) outside the domain of f(x), b) end points, c) critical points, or d) points of inflection. I don't remember obtaining any point outside this classification. This suggests to me that c is an inflection point.
     
  2. jcsd
  3. Nov 15, 2014 #2
    For Q1, by your definition, ##a## being an inflection point requires that ##f''## change sign at ##a##.

    In the event that ##f''(a)## exists, there are limitations on what can happen with ##f''## "around" ##a##. In particular, there cannot be a jump or removable discontinuity at ##a##. If ##f''(a)\neq 0## then in every interval containing ##a## there are infinitely many points ##x## with ##f''(x)## close to ##f''(a)##. It would not be possible for ##f''## to change sign at ##a## in this case. The mathematics behind all of this is a bit more advanced than what is presented in a typical intro to calculus, which based on your definition of inflection point, is the situation you're in.

    The bottom line is that if ##f''(a)## exists and ##a## is an inflection point (by your definition), then ##f''(a)=0##.

    The answer to Q2 is no. A counterexample is
    ##f(x)=x^4+x##

    Edit: See http://en.wikipedia.org/wiki/Darboux's_theorem_(analysis) for the background behind my reply to Q1. Like I said, it's a bit (not much) more advanced than what is typically presented in an introductory calculus class.
     
    Last edited: Nov 15, 2014
  4. Nov 15, 2014 #3

    LCKurtz

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    For 2, think about ##f(x) = x^4+x##.

    [Edit] Woops. I spoiled Gopher's spoiler. He must type faster...
     
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