2(a)

Let . Then . Hence g is a strictly increasing function. A strictly increasing function can only cross thex-axis at most once.

Suppose doesn’t cross thex-axis at all. The only way this is possible for a strictly increasing differentiable (and thus continuous) function is if either (i) is negative for allxand as or (ii) is positive for allxand as . In either case, we have that as . This contradicts the fact that is bounded above by . (Get it? means that gets arbitrarily close to 1 for sufficiently large values ofx; in particular, it can get within of 1, in which case, we would have the contradiction that for these sufficiently large values ofx.)

Hence g must cross the horizontal axis exactly once; in other words, f has a unique fixed point.

2(b)

For this, just choose any strictly increasing function such that (or such that ). For example, , i.e. .