Task 6: Solutions
Look at the following picture which shows f2 with various
a.
Number the curves, giving the highest curve number 1.
Which curve shows f2 with a closest to a2?
We use the information from the discussion preceding Task 6.
There we saw that at a = a2, and for some z such that f2 (z) = z, the derivative of |f2 (z)| = 1. Well, z such that f2 (z) = z, are precisely the points where the graph of y = f2 (x) meewhere the graph of y = f2 (x) meets
the line y = x. So examine, for each curve whether the modulus the
derivative of f2 is 1 - ie we are looking for the curve where
y = f2 (x) crosses y = x with a 45 degree slope.
We can rule out the first (cyan) curve, because it attains the
value 1, which only happens for a = 4. The second (purple) curve
does seem to have a (negative) 45 degree slope where it meets y = x
for the fourth time. This is the one with a closest to a2. The
third (blue) curve might also be considered a candidate. But all
the others clearly meet y = x in shallow slopes.
Which curve shows f2 with a closest to a1?
When a is less than a1, (and for x1 ¹ 0) the
population sequence converges (to a unique number). Hence the
subsequence (f2 (x1)) must also converge to a unique number.
This number is the x where f2 (x) meets the line y = x. Thus
we are looking for graphs of f2 which meet y = x in a
single point.
This is true for curves 5 (green) and 6 (red) only. The larger of
these two, 5, must have a close to a2. (In fact the 5th
(green) curve is almost tangential to y = x at the (non-zero) point
where they cross.)