maths  fixed points
posted by Anonymous .
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
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