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{\bf Question}

Sketch a graph of $f(x)=-2x^2+8x-5$, and find a partition of two
intervals from which it can be deduced that $f:\ \bf{R}
\longrightarrow \bf{R}$ has periodic points of every period.

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{\bf Answer}

Maximum of $-2x^2+3x-5$ occurs where $-4x+8=0$, i.e. $x=2$: then
$y=3$. When $x=3$ we have $y=1$; the other solution to $y=1$ is:
$x=1$.  Hence $f$ maps the interval [1,3] to itself, with max(=3)
at $x=2$. 

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\epsfig{file=314-3-2.eps, width=30mm}
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Partition $\{L,R\}$ has incidence graph
\epsfig{file=314-3-1.eps, width=20mm} and so
we have periodic orbits of every period.


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