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  3. just (c) from the problem below: 1.11. let f be a

Just (c) from the problem below: 1.11. Let f be a one-to-one smooth map of the...

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Advance Math

Just (c) from the problem below:

1.11. Let f be a one-to-one smooth map of the real lineto itself. One-to-one means that if f(xi) = f(x2), then x1 = x2. Afunction f is called increasing if x1 ? x2 implies

f(x1 )?f(x2), and decreasing if x1 ? x2 implies f(x1 ) ?f(x2 )

(a) Show that f is increasing for all x or f isdecreasing for all:x

(b)Show that every orbit {x0, x1, x2…} of f2 satisfieseither xo

x1x2… or xox1x2...

(c) Show that every orbit of f2 either diverges to+

or- or converges to a

fixed point of f2.

(d) What does this imply about convergence of the orbitsof f?

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