How can I determine if a point on a curve is truly an inflexion point?

  • Thread starter daster
  • Start date
  • Tags
    Points
In summary, when finding the point of inflexion of a curve y=f(x) where there is no x such that y'=0, you may need to use the inverse of x to find dx/dy and set it to zero. However, this may not always yield suitable values for y. It is important to understand the definition of an inflexion point, which is a point where the second derivative changes sign, and to check the behavior of the second derivative around points where f'(x)=f''(x)=0. The statement that "If f'(x)=0 and f''(x)=0, the turning point is a point of inflexion" is not always true, as shown by counterexamples.
  • #1
daster
Generally speaking, how do I find the point of inflexion of a curve y=f(x) if there is no x such that y'=0.

For example, say we have the curve y=arcsinh(x+1) and want to find its point of inflexion, so y'=1/[1+sqrt{1+(x+1)^2}]=0, but there are no values of x that do that. I tried to use the inverse of x and find dx/dy instead, then set it to zero, but again, I couldn't find a suitable value for y.

How do I approach questions like this?
 
Physics news on Phys.org
  • #2
Do you know what an inflexion point is?My guess is that u don't...So how about read and then tackle problems...??

Daniel.
 
  • #3
Set [tex] \frac{d^2y}{dx^2} = 0 [/tex]
 
  • #4
Do you know what an inflexion point is?My guess is that u don't...So how about read and then tackle problems...??

Daniel.
...

What do you think I'm doing?
 
  • #5
You weren't after the inflexion points...You were determining the critical points which is something totally different.

Daniel.
 
  • #6
My book says:
If f'(x)=0 and f''(x)=0, the turning point is a point of inflexion.

So I did that, but I can't find values of x that make f'(x) and f''(x) equal to zero...
 
  • #7
Wait, nevermind, the problem was with my differentiation.
 
  • #8
daster said:
My book says:
If f'(x)=0 and f''(x)=0, the turning point is a point of inflexion.

You asked about INFLEXION POINTS...Not about turning points...
The definition of an inflexion point is that of a point x* for which THE SECOND DERIVATIVE COMPUTED IN THAT POINT IS ZERO AND MOREOVER IT CHANGES THE SIGN...(plus-->minus or viceversa).

Daniel.
 
Last edited:
  • #9
daster said:
My book says:
If f'(x)=0 and f''(x)=0, the turning point is a point of inflexion.

So I did that, but I can't find values of x that make f'(x) and f''(x) equal to zero...
Just for the original poster's benefit, this isn't necessarily true. An inflection point is where the graph of a function changes concavity (measured by the sign of the second derivative). You should always check the behavior of the second derivative around points where f'(x)=f''(x)=0 in order to make sure that concavity changes, as it's naive to assume the second derivative (or any function) changes sign whenever the function hits 0.
For example, the function f(x)=x^4 has f''(0)=f'(0)=0, but (0,0) is not a point of inflection. It's just very, very flat. :wink: As dexter said, I also disagree with the wording of the statement. Points where f'(x)=0 should be called critical points, not turning points.
If your statement above was indicative of the wording of your book and this is all your book has to say about points of inflection, I would consider learning from another source, or complaining to the professor.
 
Last edited:
  • #10
daster said:
My book says:
If f'(x)=0 and f''(x)=0, the turning point is a point of inflexion.

So I did that, but I can't find values of x that make f'(x) and f''(x) equal to zero...

That does NOT say that f'(x) and f"(x) must both be zero for x to be an inflexion point. It says that IF f'(x)= 0 and f"(x)= 0 then the point is BOTH a turning point and a point of inflexion. i.e. it is a turning point AND a point of inflexion.
 
  • #11
HallsofIvy said:
That does NOT say that f'(x) and f"(x) must both be zero for x to be an inflexion point. It says that IF f'(x)= 0 and f"(x)= 0 then the point is BOTH a turning point and a point of inflexion. i.e. it is a turning point AND a point of inflexion.
Hi Halls,
The statement in the book is still false (as shown above). The book may be defining "point of inflection" this way, but it doesn't fit any other definition I've seen and has no unique graphical meaning beyond "Well, f''(x)=0 at that point.". :smile:
To the original poster, remember that a point of inflection is a point where the second derivative (concavity) changes sign. Study the behavior of the second derivative to see whether there are any such points. As HallsofIvy said, the first derivative need not be 0 at a point of inflection. The function f(x)=cos(x) has a point of inflection at x=pi/2, for example.
 
Last edited:
  • #12
Thank you. That occurred to me but I didn't say it. f"= 0 is a necessary condition but not sufficient! If y= x4 then f"(x)= 12x2.
The second derivative is 0 at x= 0 but does not change sign there. (0,0) is NOT an inflexion point.
 

What are points of inflexion?

Points of inflexion are points on a curve where the concavity changes. This means that the curve goes from being convex (curving upwards) to concave (curving downwards) or vice versa. These points are important in mathematical analysis and can help determine the behavior of a function.

How do you find points of inflexion?

To find points of inflexion, you need to take the second derivative of the function and set it equal to 0. Solve for x, and these values will be the x-coordinates of the points of inflexion. Then, plug these values into the original function to find the corresponding y-coordinates.

Why are points of inflexion important?

Points of inflexion can help us understand the shape and behavior of a function. They can also indicate when the function changes direction, which can be useful in real-world applications such as predicting the maximum or minimum value of a variable.

Can a function have more than one point of inflexion?

Yes, a function can have multiple points of inflexion. This occurs when the concavity of the curve changes multiple times. In fact, some functions may have an infinite number of points of inflexion, such as the function y = x^3.

Are points of inflexion the same as critical points?

No, points of inflexion and critical points are not the same. Critical points are points where the derivative of a function is equal to 0 or undefined, while points of inflexion are points where the second derivative is equal to 0. However, a critical point can also be a point of inflexion if the second derivative changes sign at that point.

Similar threads

Replies
1
Views
1K
Replies
12
Views
1K
Replies
3
Views
172
Replies
20
Views
2K
  • Calculus
Replies
5
Views
1K
  • Other Physics Topics
Replies
4
Views
10K
Replies
2
Views
2K
Replies
1
Views
751
Back
Top