Finding f concavity and inflection points from a graph of f'

[sjaguar13]

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?

 

[arildno]

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"?

 

[sjaguar13]

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.

 

[arildno]

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?

 

[sjaguar13]

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).

 

[Data]

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?

 

[sjaguar13]

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.

 

[Data]

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?