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SafiBTA

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## Homework Statement

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?

## Homework Equations

, theorems, and definitions[/B]- 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.

## 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.