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

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SUMMARY

The discussion focuses on the relationship between a function \( f \) and its derivative \( f' \), specifically regarding intervals of increase and decrease, concavity, and inflection points. It is established that if \( f' \) is decreasing, \( f \) may not necessarily be decreasing in the same interval, as demonstrated with the function \( f = C - x^2 \). The mathematical definitions clarify that \( f \) is decreasing when \( f' < 0 \) and that concavity is determined by the sign of the second derivative \( f'' \). Inflection points occur where the concavity changes, specifically at the endpoints of intervals of concavity.

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  • Understanding of derivatives and their significance in calculus.
  • Knowledge of concavity and inflection points in functions.
  • Familiarity with critical points and their classification (maximum, minimum, inflection).
  • Ability to analyze graphs of functions and their derivatives.
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  • Study the concept of second derivatives and their role in determining concavity.
  • Learn how to identify and classify critical points using the first and second derivative tests.
  • Explore the graphical interpretation of functions and their derivatives to enhance understanding of increasing/decreasing behavior.
  • Investigate the implications of inflection points in real-world applications and their significance in function analysis.
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Students and educators in calculus, mathematicians analyzing function behavior, and anyone seeking to deepen their understanding of derivatives, concavity, and critical points in mathematical functions.

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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?
 
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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 f=C-x^{2}
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 x-1 is so important? Let's say my function is

f(x) = \sin{(\pi x)}

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

f(x) = \sin{(\pi x)}

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


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

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

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