= Solution
Stationarity gives the conserved <Killing energy> per unit rest <mass> $E=(1+x^2)\dot t$, where dots here denote proper-time <derivatives>. It measures <energy> relative to the time-translation Killing field; a static local observer measures $E/\sqrt{1+x^2}$. Timelike normalization is
$$
\frac{\dot x^2}{1+x^2}-(1+x^2)\dot t^2=-1.
$$
Eliminating $\dot t$ and setting $a^2=E^2-1$ gives
$$
\boxed{\dot t=\frac{E}{1+x^2},\qquad\dot x=\pm\sqrt{a^2-x^2}.}
$$
Thus, on each increasing/decreasing leg respectively, $d\tau=\pm dx/\sqrt{a^2-x^2}$ and $dt=\pm E\,dx/[(1+x^2)\sqrt{a^2-x^2}]$. The printed positive <differentials> describe the increasing leg, not both legs at once. Differentiation along a leg gives $\ddot x=-x$, extending smoothly through the turning points. Hence $x=a\sin(\tau-\tau_0)$ and \b[the proper-time period is $2\pi$].
The requested <velocity> sketches are the branches $\dot x=\pm\sqrt{a^2-x^2}$ and $dx/dt=\pm(1+x^2)\sqrt{a^2-x^2}/E$. They share the turning points and opposite signs on the two legs; their speed ratio is $(1+x^2)/E$, smaller than one near zero and larger than one near the endpoints. The supplied <integral> is the coordinate time from $-a$ to $a$, equal to $\pi$. The return leg takes the same time, so \b[the coordinate-time period is also $2\pi$].
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ii/paper-1-geodesic-speeds.png]
{title=Proper-time and coordinate-time velocity branches of the bounded timelike geodesic}
For the perturbed metric the same Killing conservation gives $\dot t=E/(1+x^2)$, and normalization becomes $(1+f(x))^2\dot x^2=a^2-x^2$. Since $1+f>0$,
$$
\boxed{d\tau=\pm\frac{1+f(x)}{\sqrt{a^2-x^2}}\,dx,\qquad dt=\pm\frac{E[1+f(x)]}{(1+x^2)\sqrt{a^2-x^2}}\,dx.}
$$
Both weights without $f$ are even in $x$. Oddness of $f$ makes its <integral> over the symmetric excursion $[-a,a]$ vanish. Each increasing leg therefore still takes $\pi$ in either clock; the decreasing leg takes the same <integrals> with reversed orientation. \b[Both periods remain exactly $2\pi$], although the proper-time waveform is no longer generally sinusoidal. This is the <odd radial perturbations preserve timelike-geodesic periods> mechanism. Smoothness of the metric and positivity of $1+f$ ensure regular turning and monotone coordinate time.
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