Queries regarding Inflection Points in Curve Sketching

[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.

 

[gopher_p]

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

Spoiler

$f(x)=x^4+x$

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.

 

[LCKurtz]

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 $f(x) = x^4+x$.

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