= Solution
Put $F(N)=f(N,1/2)$. A <cobweb plot> alternates a vertical move to the <function graph> $y=F(N)$ with a horizontal move to the diagonal $y=N$. Beginning at $N_0=3/2$, the first <iterates> are
$$
N_1=\frac34,\qquad N_2=\frac{27}{64}\simeq0.421875,\qquad N_3\simeq0.571278.
$$
The diagram shows the ensuing approach to $1/2$ with alternating deviations. More precisely, if $e_t=N_t-1/2$, <Taylor expansion> at the <fixed point> gives
$$
e_{t+1}=-\frac34e_t+O(e_t^2).
$$
For sufficiently small nonzero deviations the sign reverses each step and their magnitudes contract. \b[The local approach is an alternating, geometrically damped oscillation], with asymptotic contraction factor $3/4$ per iterate.
\Image[/past-exam-of-the-topics-in-applied-mathematics-course-of-the-university-of-oxford/2008-cubic-cobweb.png]
{title=Cobweb approaching the attracting fixed point}
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