Is a Monotonic Function Always Increasing at x=0?

  • Thread starter player1_1_1
  • Start date
  • Tags
    Function
In summary, the conversation discusses the concept of increasing functions and how it relates to the theorem that states a function is increasing when its derivative is positive. The definition of an increasing function is also mentioned, which does not make sense to talk about at a specific point. It is also stated that y=x^3 is increasing on any interval.
  • #1
player1_1_1
114
0

Homework Statement


[tex]y=x^3[/tex]

The Attempt at a Solution


I know that function is increasing when [tex]f'(x) > 0[/tex] but in [tex]x=0[/tex] there is [tex]f'(x) = 0[/tex], so is function increasing there or not? From definition I know that its increasing there, but how can I connect this with theorem that function is increasing when [tex]f'(x) > 0[/tex]?
 
Physics news on Phys.org
  • #2
What precisely is your definition of an increasing function?
 
  • #3
[tex]\forall_{x_1,x_2\in X} \left( x_1 < x_2 \Rightarrow f(x_1)<f(x_2) \right)[/tex]
 
  • #4
If a function is non-decreasing (weakly increasing), and [tex]f(x_1)=f(x_2)[/tex], [tex]x_1\neq x_2[/tex], then f is constant on [tex][x_1, x_2][/tex], so a single zero of a derivative cannot spoil injectivity.
 
  • #5
aha, so if derivative is positive and 0 only for countable set of points the function will be increasing?
 
  • #6
Yes. It can probably be strenghtened a little bit, but that should be enough for all practical purpose.
 
  • #7
player1_1_1 said:
[tex]\forall_{x_1,x_2\in X} \left( x_1 < x_2 \Rightarrow f(x_1)<f(x_2) \right)[/tex]
With that definition, it makes no sense to talk about a function being "increasing" at a point. It is easy to prove that [itex]y= x^3[/itex] is increasing on any interval.
 
  • #8
thanks for answers. Is any definition which can define increasing in a point?
 
  • #9
stupid question, sorry, nvm
 

1. What is a monotonic function?

A monotonic function is a mathematical function that either always increases or always decreases as its input variable increases. In other words, the function maintains a consistent direction, either going up or going down, with no sudden changes in direction.

2. Is a monotonic function the same as an increasing function?

No, a monotonic function can either be increasing or decreasing, while an increasing function only moves in a positive direction. A monotonic function can also have negative values, as long as it maintains the same direction.

3. What does "x=0" mean in relation to a monotonic function?

The "x=0" in the question refers to the value of the input variable, also known as the independent variable, in the function. It is the point at which we are examining the behavior of the function to determine if it is increasing or decreasing.

4. Can a monotonic function ever change direction?

No, a monotonic function cannot change direction. It must maintain a consistent direction, either increasing or decreasing, throughout its entire domain.

5. Are all monotonic functions continuous?

Not necessarily. A monotonic function can have discontinuities, such as jumps or asymptotes, as long as it maintains the same direction. However, a continuous monotonic function is also possible, where there are no sudden breaks or interruptions in the function.

Similar threads

  • Calculus and Beyond Homework Help
Replies
10
Views
815
  • Calculus and Beyond Homework Help
Replies
8
Views
302
  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
131
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
26
Views
804
  • Calculus and Beyond Homework Help
Replies
3
Views
764
  • Calculus and Beyond Homework Help
Replies
4
Views
614
  • Calculus and Beyond Homework Help
Replies
2
Views
370
Back
Top