Consider the following figure fix 35x1x a For rx kx1x the v

Consider the following figure. fix)- 3.5x(1-x (a) For rx) = kx(1-x), the values of a and b for which f(a)-b and b) = a are given by the formulas 1+k+(k- 3)(k1) 2k and 1 k-(k-3)(k 1) 2k Let(x)-3.5x(1-x). Use the formulas above to find values of a and b such that [a, b] is a 2-cycle for this function [that is, so that na)-b andb)-a]. Exact answers are required, not calculator approximations. a= 6/7 b= 3/7 (b) The figure above shows the iteration process for the 2-cycle determined in part (a). Use the answers in part (a) to specify the coordinates of the four points P, Q, R, and S. (x, y) = (x, y) = (x, y) = (x, y) = Need Help? S Talk to a Tutor

Solution

a=6/7, b=3/7

Point P is the point that lies on line y=x and is lower than Q. Hence

P=(b,b)= (3/7,3/7)

X coordinate of P and Q is the same. hence we find the corresponding value of y moving from P to Q

y(b)=a ----> y(3/7)=6/7

Q=(b,y(b))=(b,a)= (3/7,6/7)

R lies on line y=x.

R=(a,a)=(6/7,6/7)

S lies on the curve and corresponds to x value same as that of R.

S=(a,y(a))=(a,b)=(6/7,3/7)