Sine: Does it Have a Point of Inflection?

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Discussion Overview

The discussion revolves around the concept of inflection points in the context of the sine function and related polynomial functions. Participants explore definitions and conditions for inflection points, including the role of the first and second derivatives, and whether sine has inflection points at specific values.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether sine has a point of inflection at π/2, citing that the second derivative changes sign there, while the first derivative is not zero.
  • Others argue that an inflection point is defined by a change in concavity, which may not necessarily require the first derivative to be zero.
  • There is a discussion about the function f(x) = x^4, where participants note that its second derivative is zero at x=0, but it does not represent an inflection point due to the lack of a change in concavity.
  • Some participants mention that straight lines have zero second derivatives and no inflection points, while polynomial functions of even degree (like x^4) exhibit peculiar behavior regarding inflection points.
  • Conditions for inflection points are discussed, including the necessity of a change in concavity and the order of the lowest non-zero derivative, with some uncertainty about the sufficiency of these conditions.

Areas of Agreement / Disagreement

Participants express differing views on the conditions for inflection points, particularly regarding the necessity of the first derivative being zero. There is no consensus on whether sine has inflection points at specific values or on the implications of the second derivative being zero for certain functions.

Contextual Notes

Limitations include varying interpretations of inflection points, dependence on definitions of concavity, and unresolved mathematical steps regarding the behavior of specific functions.

FedEx
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Hi

Does sine have a point of inflexion?

Well there are many interpretations for the point if infelxion...

I saw on wiki that sin2x has point of inflexion. Well the second derivatie does change its sign at pi by 2, but the first derivative is not eqaul to zero. Can we say that sine has a point if infelxion at pi by two. WIKI says yes...
 
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it has nothing to do with the first derivative. an inflection point is where the second derivative = 0 so sin(x) has inflection pts at integer multiples of pi
 
fourier jr said:
it has nothing to do with the first derivative. an inflection point is where the second derivative = 0 so sin(x) has inflection pts at integer multiples of pi

Untrue,consider f(x) = x4. f ''(x) = 12x2. f '' (0) = 0, but that is not an inflection point.

Inflection point is when the concavity changes.
 
l'Hôpital said:
Untrue,consider f(x) = x4. f ''(x) = 12x2. f '' (0) = 0, but that is not an inflection point.

Inflection point is when the concavity changes.

When the function is concave up it has a positive second derivative and when it is concave down it has a negative second derivative. Thus, at the point where the function switches from concave up to concave down (the inflection point) the second derivative would be zero. However, as you point out, some functions have points that have second derivatives that are zero but they are not inflection points, such as straight lines. It seems the function x^4 is one of these odd exceptions. I have not realized this before. Is the function x^4 briefly a flat line at x=0?

Straight lines obviously have second derivatives that are zero but have no points of inflection. f(x) = x^4 is strange though. Does it have a region around x=0 where it is perfectly straight? The same 'problem' occurs for x^6, x^8, etc...not for x^2 though. The problem doesn't occur for x^3, x^5, x^7, etc since x=0 is indeed an inflection point for these functions.
 
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l'Hôpital said:
Untrue,consider f(x) = x4. f ''(x) = 12x2. f '' (0) = 0, but that is not an inflection point.

Inflection point is when the concavity changes.
deleted
 
leright said:
Straight lines obviously have second derivatives that are zero but have no points of inflection. f(x) = x^4 is strange though. Does it have a region around x=0 where it is perfectly straight? The same 'problem' occurs for x^6, x^8, etc...not for x^2 though. The problem doesn't occur for x^3, x^5, x^7, etc since x=0 is indeed an inflection point for these functions.

f(x) = ax2n is always a parabola for any positive integer n, so concavity is always the same sign of a.
 
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l'Hôpital said:
f(x) = ax2n is always a parabola for any positive integer n, so concavity is always the same sign of a.

I see that. This is very strange to me.
 
leright said:
I see that. This is very strange to me.

Why?

You believe y = x2 is a parabola.

Let x = un, for some positive integer n.

then y = u2n. Still a parabola. : )

You can think of it in quadratic form.

As long as you have y = ax2n+ bxn + c, you have a parabola.
 
l'Hôpital said:
Why?

You believe y = x2 is a parabola.

Let x = un, for some positive integer n.

then y = u2n. Still a parabola. : )

You can think of it in quadratic form.

As long as you have y = ax2n+ bxn + c, you have a parabola.

I don't think it is strange that it is a parabola. I think it is strange that its second derivative is zero at x=0.
 
  • #10
l'Hôpital said:
Why?

You believe y = x2 is a parabola.

Let x = un, for some positive integer n.

then y = u2n. Still a parabola. : )

You can think of it in quadratic form.

As long as you have y = ax2n+ bxn + c, you have a parabola.

A parabola only if we plot y versus x^n(ofcourse)
 
  • #11
Well let me write down all the conditions which i know for a point to be a point if inflection

1)Concavity changes. That is if concave upwards than changes to concave downwards

2)It should be a critical point. dy/dx = zero.

3) The lowest order non zero derivative should be of odd order( ie third,fifth...)

I don't which of them is the necessary and the sufficient condition
 
  • #12
FedEx said:
Well let me write down all the conditions which i know for a point to be a point if inflection

1)Concavity changes. That is if concave upwards than changes to concave downwards

2)It should be a critical point. dy/dx = zero.

3) The lowest order non zero derivative should be of odd order( ie third,fifth...)

I don't which of them is the necessary and the sufficient condition

a point doesn't need to be a critical point to be an inflection point.
 
  • #13
Nicely salved ur equation...
fedextracking.org[/URL][/color][/right]​
 
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