Sine: Does it Have a Point of Inflection?

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SUMMARY

The sine function, sin(x), has points of inflection at integer multiples of π, where its second derivative equals zero. The discussion clarifies that while the first derivative does not need to be zero at these points, the change in concavity is the defining characteristic of an inflection point. The example of f(x) = x^4 illustrates that a zero second derivative does not guarantee an inflection point, as it does not change concavity at x=0. The conditions for identifying inflection points include changes in concavity and the order of the lowest non-zero derivative.

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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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