Problem
of the
Week

PROBLEM 72


The following problem was suggested by Yan Fridman.  Many thanks!

Let  f1(x) = 4x(1-x), for x on the interval between 0 and 1.  The graph is given on the left.  Define  f2(x) = f(f(x)),  f3(x) = f(f(f(x)), and so on.  Note that

f1(3/4) = f2(3/4) = f3(3/4) = ...,

and that, similarly,

f2(1/4) = f3(1/4) = f4(1/4) = ...

Find all values of x between 0 and 1 which become eventually stationary; that is, for which, for n sufficiently large,

fn(x) = fn+1(x) = fn+2(x) = ...

(Hint:  Consider the graphs of  y = f(x)  and  y = x.)


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