Queries regarding Inflection Points in Curve Sketching

• SafiBTA
In summary, the set D, which is the union of the set of roots and the set of points where the second derivative does not exist, contains all the potential inflection points of the function f(x). There cannot exist any inflection points of f(x) outside of this set. Additionally, if a point c is in D but not in the set of critical points A, it is not necessarily an inflection point.
SafiBTA

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.

SafiBTA said:

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.

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:
SafiBTA said:

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.

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

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

What is an inflection point?

An inflection point is a point on a curve where the curvature changes direction. This means that the concavity of the curve changes from being concave up to concave down, or vice versa.

How do you find inflection points?

To find inflection points, you must first find the second derivative of the function. Then, set the second derivative equal to zero and solve for x. The resulting x-values are the potential inflection points. To confirm if they are actual inflection points, you can use the first derivative test to check the concavity at those points.

What is the significance of inflection points in curve sketching?

Inflection points can provide important information about the behavior of a curve. They can help determine the shape of the curve and identify any changes in direction or concavity. In addition, inflection points can also be used to find the points of inflection, which are the points where the curve changes from being concave up to concave down or vice versa.

Can a curve have more than one inflection point?

Yes, a curve can have multiple inflection points. The number of inflection points a curve has depends on the complexity and behavior of the function. For example, a cubic function can have up to two inflection points, while a higher degree polynomial can have more inflection points.

Do all functions have inflection points?

No, not all functions have inflection points. In order for a function to have inflection points, its second derivative must change sign at some point. This means that the function must have a point where the concavity changes from concave up to concave down or vice versa. For instance, a linear function does not have any inflection points since its second derivative is always zero.

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