Sine: Does it Have a Point of Inflection?

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In summary, the conversation discusses the concept of inflection points and the conditions for a point to be considered an inflection point. It is determined that the function x^4 is an exception to the rule as it has a second derivative of zero at x=0 but is not considered an inflection point. The conversation also touches on the idea of parabolas and their relationship to inflection points.
  • #1
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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  • #2
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
 
  • #3
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.
 
  • #4
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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  • #5
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.
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  • #6
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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  • #7
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.
 
  • #8
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.
 
  • #9
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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1. What is a point of inflection?

A point of inflection is a point on a curve where the direction of concavity changes. This means that the curve changes from being concave up (opening upwards) to concave down (opening downwards) or vice versa.

2. How do you find the point of inflection for a sine function?

To find the point of inflection for a sine function, you can use the second derivative test. Take the second derivative of the sine function and set it equal to 0. Solve for x to find the x-coordinate of the point of inflection. Then, plug this x-coordinate into the original sine function to find the y-coordinate of the point of inflection.

3. Can a sine function have more than one point of inflection?

Yes, a sine function can have more than one point of inflection. In fact, it can have an infinite number of points of inflection, as the curve continuously changes from being concave up to concave down.

4. What does the presence or absence of a point of inflection tell us about a sine function?

If a sine function has a point of inflection, it means that the curve is changing direction at that point. This can indicate a change in behavior or a critical point on the curve. If a sine function does not have a point of inflection, it means that the curve is continuously concave up or continuously concave down, with no changes in direction.

5. How does a point of inflection affect the graph of a sine function?

A point of inflection can affect the shape of the graph of a sine function. If the point of inflection is at the origin (0,0), the curve will be symmetrical and have a smooth, rounded shape. If the point of inflection is not at the origin, the curve may appear more angular or have a different shape. Additionally, the point of inflection can divide the curve into different sections with different rates of change.

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