Graphes: Increasing/Decreasing, Concave Up/Down, Inflection Points?

  • Context: Undergrad 
  • Thread starter Thread starter sjaguar13
  • Start date Start date
  • Tags Tags
    Concave Points
Click For Summary

Discussion Overview

The discussion revolves around the relationships between a function and its first and second derivatives, particularly focusing on increasing and decreasing intervals, concavity, and inflection points. Participants explore these concepts through both graphical and algebraic perspectives, raising questions about definitions and implications of derivative behavior.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether the intervals where the first derivative f' is decreasing correspond to the intervals where the function f is decreasing, with examples provided.
  • There is a discussion about the mathematical definition of a function being decreasing, with some participants asserting that it relates to the behavior of the derivative.
  • One participant suggests that if f' < 0, then f is decreasing, and if f' is decreasing, then f is concave down, while also discussing conditions for local maxima and minima.
  • Another participant raises a specific example involving the function f(x) = sin(πx) and questions the nature of the critical point at x = 1/2, seeking clarification on whether it represents a maximum, minimum, or inflection point.
  • There is confusion expressed by participants regarding the implications of critical points and the importance of the point x-1 in determining local extrema.

Areas of Agreement / Disagreement

Participants express differing views on the relationships between the behavior of a function and its derivatives, particularly regarding definitions and implications of decreasing intervals and critical points. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants rely on definitions and interpretations of decreasing functions, concavity, and critical points, which may vary. There are unresolved mathematical steps and assumptions regarding the behavior of derivatives and their implications.

sjaguar13
Messages
49
Reaction score
0
If I had a graph of f', would the intervals where f' is decreasing be the same as the intervals where f is decreasing? For example, the graph of f' is degreasing from -1 to 0. Is f decreasing from -1 to 0 also?

With the same graph of f', how would you know on which intervals f is concave up or down? Would I need to find f'' and how would I do that with just the graph?

Same thing with the inflection points, would I need f''?

If I didn't have a graph, just the equation of f', would I find the intervals of increasing and degreasing by finding what numbers make f' = 0, and check wether those numbers if put back into the equation come out positive or negative?
 
Physics news on Phys.org
sjaguar13 said:
If I had a graph of f', would the intervals where f' is decreasing be the same as the intervals where f is decreasing? For example, the graph of f' is degreasing from -1 to 0. Is f decreasing from -1 to 0 also?
No, let [tex]f=C-x^{2}[/tex]
f is increasing on [-1,0], whereas f' is decreasing there.
What does it mean that a function is decreasing?
What is the mathematical definition of "decreasing"?
 
What does it mean that a function is decreasing? From left to right, x is getting smaller

What is the mathematical definition of "decreasing"? I assume the same thing.
 
Let's take the case of a differentiable function:
Given a function f(x), what is the sign of its derivative, f'(x), when f is decreasing?
 
I think I got this. If f'<0, f is decreasing. If f' is decreasing, f is concave down. If f'(x) = 0, then x is either a local max or min. It's min if f'(x-1) < 0 and max if f'(x-1) > 0. The inflection points will be where the direction of concavity changes, so it's going to be the endpoints of the intervals, like concave up on (2,5)U(7,10) and down on (0,2)U(5,7) then inflection points would be 2,5,7 (not 2 and 10 because the are end points).
 
It's min if f'(x-1) < 0 and max if f'(x-1) > 0.

Why do you think the point [tex]x-1[/tex] is so important? Let's say my function is

[tex]f(x) = \sin{(\pi x)}[/tex]

does the critical point at [tex]x = \frac{1}{2}[/tex] represent a minimum, maximum, or point of inflection, according to your method? What is it really?
 
Data said:
Why do you think the point [tex]x-1[/tex] is so important? Let's say my function is

[tex]f(x) = \sin{(\pi x)}[/tex]

does the critical point at [tex]x = \frac{1}{2}[/tex] represent a minimum, maximum, or point of inflection, according to your method? What is it really?


...I'm confused.
 
Do you agree that [tex]\sin (\pi x)[/tex] has a critical point (ie. a point where [tex]f^\prime = 0[/tex]) at [tex]x = \frac{1}{2}[/tex]?

If so, all I want to know is: does this critical point represent a maximum, a minimum, or a point of inflection?
 

Similar threads

MHB Is f(x)=ln(x)/x Increasing or Decreasing?
  • · Replies 1 ·
Replies
1
Views
3K
MHB Determine the relative maximum and minimum on the graph
  • · Replies 5 ·
Replies
5
Views
2K
High School Second derivatives and inflection points
  • · Replies 5 ·
Replies
5
Views
4K
MHB 141.30 how many points of inflection will the graph of the function have
  • · Replies 3 ·
Replies
3
Views
2K
MHB A Function's Intervals of Increasing/Decreasing, Extrema and Concavity
  • · Replies 25 ·
Replies
25
Views
6K
MHB Is the Inflection Point at x = 7 Incorrect?
  • · Replies 5 ·
Replies
5
Views
2K
High School Why is Big-O about how rapidly the Taylor graph approaches that of f(x)?
  • · Replies 19 ·
Replies
19
Views
4K
Undergrad What can we learn from higher derivatives in calculus?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad The Second derivative test for Concavity
  • · Replies 9 ·
Replies
9
Views
3K
MHB Concave Up or Down? Identifying Inflection Points in a Polynomial Function
  • · Replies 4 ·
Replies
4
Views
2K