Inflection point of non continuous or non differentiable function

In summary, the conversation discusses three functions and their potential inflection points. The first two functions do not meet the definition of an inflection point because they are not continuous at the point of potential inflection. The third function may qualify as an inflection point due to a change in concavity at the point, but it ultimately depends on the definition being used.
  • #1
player1_1_1
114
0

Homework Statement


three functions:
[tex]y=\begin{cases}\arctan \frac{1}{x}\ x\neq0\\ 0\ x=0\end{cases}[/tex]
[tex]y=\frac{1}{x}, y=|x^2-1|[/tex] and what about inflection point?

The Attempt at a Solution


first function is concave on left of 0, convex on right, so from definition it should be inflection point, but its not continuous in this point, a function need to be continuous in this place or not?
in 2, [tex]x=0[/tex] should be inflection point, but its not in the domain, so is there inflection point?
in 3, function is continuous in [tex]x=1[/tex] but not differentiable, is there inflection point or not?
 
Physics news on Phys.org
  • #2
up,.
 
  • #3
player1_1_1 said:

Homework Statement


three functions:
[tex]y=\begin{cases}\arctan \frac{1}{x}\ x\neq0\\ 0\ x=0\end{cases}[/tex]
[tex]y=\frac{1}{x}, y=|x^2-1|[/tex] and what about inflection point?

The Attempt at a Solution


first function is concave on left of 0, convex on right, so from definition it should be inflection point, but its not continuous in this point, a function need to be continuous in this place or not?
in 2, [tex]x=0[/tex] should be inflection point, but its not in the domain, so is there inflection point?
in 3, function is continuous in [tex]x=1[/tex] but not differentiable, is there inflection point or not?

It probably depends on the definition your text gives. Most say it must be a point on the graph where the concavity changes. That would rule out the first two. I would say the third qualifies because of the change in concavity at the point. But your mileage may vary.
 
  • #4
thx!
 

What is an inflection point?

An inflection point is a point on a graph where the rate of change of a function changes from increasing to decreasing, or vice versa. It is also where the concavity of the function changes.

Can a non-continuous function have an inflection point?

Yes, a non-continuous function can have an inflection point. The function may not be defined at the inflection point, but it can still exist.

Can a non-differentiable function have an inflection point?

Yes, a non-differentiable function can have an inflection point. The function may not have a derivative at the inflection point, but it can still exist.

How is an inflection point different from a critical point?

An inflection point is where the concavity of a function changes, while a critical point is where the derivative of a function is equal to zero or undefined. Not all critical points are inflection points, and not all inflection points are critical points.

How can you find the inflection point of a non-continuous or non-differentiable function?

The inflection point of a non-continuous or non-differentiable function can be found by looking for points where the concavity changes. This can be done by finding the second derivative of the function and setting it equal to zero to find the points where the concavity changes. However, it is important to note that not all inflection points can be found using this method, as some may be undefined due to the function being non-differentiable.

Similar threads

Replies
1
Views
440
  • Calculus and Beyond Homework Help
Replies
5
Views
943
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
221
  • Calculus and Beyond Homework Help
Replies
4
Views
930
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
170
  • Calculus and Beyond Homework Help
Replies
3
Views
116
  • Calculus and Beyond Homework Help
Replies
1
Views
170
  • Calculus and Beyond Homework Help
Replies
27
Views
606
Back
Top