= Solution
For the specified flat matter-vacuum model,
$$
F(z)=s-\sqrt{0.3s^3+0.7}.
$$
Its small-<redshift> behaviour is $F(z)=0.55z+O(z^2)$, since $q_0=\Omega_{m,0}/2-\Omega_{\Lambda,0}=-0.55$. At high <redshift>, matter dominates and $F\sim-\sqrt{0.3}s^{3/2}$, so the initially positive <redshift drift> must turn negative. To locate the crossing, square $s=\sqrt{0.3s^3+0.7}$; both sides are positive, so squaring introduces no positive-$s$ spurious root. Factorization gives
$$
0.3s^3-s^2+0.7=(s-1)(0.3s^2-0.7s-0.7).
$$
Besides $z=0$, the physical <nonzero redshift-drift root in flat matter-Lambda cosmology> is
$$
\boxed{z_* =\frac{1+\sqrt{133}}6\simeq2.089.}
$$
The <redshift drift> is positive for $0<z<z_*$ and negative for $z>z_*$. Its maximum satisfies $1=0.45s^2/\sqrt{0.3s^3+0.7}$ and lies near $z=0.949$, where $F\simeq0.240$. This zero is distinct from the acceleration transition at $z_{\rm acc}=(2\Omega_{\Lambda,0}/\Omega_{m,0})^{1/3}-1\simeq0.671$: <redshift drift> compares expansion at two epochs, while acceleration is local to one epoch.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-72-redshift-drift.png]
{title=Redshift drift in three flat cosmologies, with the positive-to-negative crossing of the matter-vacuum model}
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