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

In summary, the conversation discusses the relationship between the intervals where the derivative of a function is decreasing and where the function itself is decreasing. It also explores how to determine the concavity and inflection points of a function using its derivative. Additionally, the conversation touches on the mathematical definitions of decreasing and critical points, as well as the significance of critical points in determining maximums, minimums, and points of inflection.
  • #1
sjaguar13
49
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 postive or negative?
 
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  • #2
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"?
 
  • #3
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.
 
  • #4
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?
 
  • #5
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).
 
  • #6
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?
 
  • #7
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.
 
  • #8
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?
 

1. What is the definition of an increasing graph?

An increasing graph is a graph in which the values of the dependent variable (usually denoted by y) increase as the values of the independent variable (usually denoted by x) increase.

2. How do you determine if a graph is decreasing?

A graph is decreasing if the values of the dependent variable decrease as the values of the independent variable increase. In other words, as x increases, y decreases.

3. What does it mean for a graph to be concave up?

A graph is concave up when the slope of the graph is increasing. In other words, the graph is curving upwards.

4. How can you tell if a graph is concave down?

If the slope of a graph is decreasing, the graph is concave down. This means that the graph is curving downwards.

5. What is an inflection point on a graph?

An inflection point is a point on a graph where the concavity changes from concave up to concave down, or vice versa. It is where the slope of the graph changes from increasing to decreasing, or vice versa.

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