Solve f(x): Find Attracting Interval w/ Gradient Criterion

[feely]

1. In this question, f is the function f(x) = 1/4x^2 - 1/8x - 5/8

The first part of the question asked to find the fixed points, which I got to 3.65 and 0.85, and to classify them as attracting, repelling or indifferent, which I did, however, the second part asks

2. Use the gradient criterion to determine an interval of attraction for one of the fixed points of f.


I haven't got an attempt at this to show, because I don't really understand the process, and that's what I'm hoping to get help with. I'm not looking the answer, more the process. That way, hopfully I will be able to answer it myself in future.

Thanks in advance.

Sean

PS - I really hope I have posted this in the correct place, if not, I'm very sorry.

 

[HallsofIvy]

1. In this question, f is the function f(x) = 1/4x^2 - 1/8x - 5/8

The first part of the question asked to find the fixed points, which I got to 3.65 and 0.85, and to classify them as attracting, repelling or indifferent, which I did, however, the second part asks

2. Use the gradient criterion to determine an interval of attraction for one of the fixed points of f.


I haven't got an attempt at this to show, because I don't really understand the process, and that's what I'm hoping to get help with. I'm not looking the answer, more the process. That way, hopfully I will be able to answer it myself in future.

Thanks in advance.

Sean

PS - I really hope I have posted this in the correct place, if not, I'm very sorry.

An "interval of attraction" of a point is an interval in which the "motion" tends back to the point. In this case, the "motion" is given by the derivative. If the derivative is negative, the "motion" is decreasing, if the derivative is positive, increasing.

If, as you say, the fixed points are 0.85 and 3.65,(I didn't check that myself) then an interval of attraction for 0.85 is all x< 0.85 where the derivative is positive (so the "motion" is toward 0.85) and all 0.85< x< 3.65 where the derivative is negative. An interval of attraction for 3.65 is all 0.85< x< 3.65 where the derivative is positive and all x> 3.65 where the derivative in negative.

 

[seboastien]

you american really have a weird way of wording your explanations

 

[berkeman]

you american really have a weird way of wording your explanations

I found Halls' explanation to be quite good. But then again, I am an American