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.