Fixed points of f?
f(x) = 1/4x^2 - 1/8x - 5/8
a) use algebra to find the fixed points and classify them.
b) use the gradient criterion to determine an interval of attraction for one of the fixed points of f
c) find exact values of the 2nd and 3rd terms of the sequence xn obtained by iterating f with initial term x0= -3.
State the long term behaviour of this sequence
My answers;
a) fixed points 5 and -1/2
f '(5) = 1/2 x 5 - 1/8 = 19/8 >1 therefore it is repelling
f '(-1/2) = 1/2 x -1/2 - 1/8 = -3/8 <1 and is therefore attracting
b) -1 < 1/2x - 1/8 < 1
So, 1/2x - 1/8 < 1 = x < 9/4
And, -1 < 1/2x - 1/8 = -7/4 < x
thus, -7/4 < x < 9/4
so, interval of attracting for point -1/2 is (-7/4 , 3/4)
c) x0 = -3
x1 = (1/4 x -3^2) - (1/8 x -3) - 5/8
= 2
x2 = (1/4 x 2^2) - (1/8 x 2) - 5/8
= 1/8
long term behaviour is;
xn tends to -1/2 as n tends to infinity.
If someone could check my answers I would be very grateful. Thanks
Your answers are mostly correct, but there are a couple of mistakes:
a) You correctly found the fixed points to be 5 and -1/2. However, you made a mistake in classifying them. The derivative at x = 5 is not 19/8, but rather 5/8. Since 5/8 is less than 1, it is an attracting fixed point. Similarly, the derivative at x = -1/2 is -3/8, which is not less than 1, so it is also a repelling fixed point.
b) Your interval of attraction for the fixed point -1/2 is not correct. The correct inequality should be: -1/2 < (1/2)x - (1/8) < 1.
Simplifying this inequality, we get: -7/4 < x < 3/4.
So the interval of attraction for the fixed point -1/2 is (-7/4, 3/4).
c) For x0 = -3, you correctly found x1 = 2. However, you made a mistake in calculating x2. The correct calculation should be:
x2 = (1/4)(2^2) - (1/8)(2) - 5/8 = 1/2 - 1/4 - 5/8 = -5/8.
So the correct exact values of the 2nd and 3rd terms of the sequence xn are: x1 = 2 and x2 = -5/8.
The long-term behavior of the sequence xn tends to -1/2 as n tends to infinity. This is because the fixed point -1/2 is attracting, so as we repeatedly apply the function f, the values of xn will approach -1/2.