= Solution
The stationary-point relation is $\Omega_{m,0}a_0^3/a_{\min}=2\Omega_{\Lambda,0}a_{\min}^2$. Consequently
$$
\dot a_{\min}^2=H_0^2\left(3\Omega_{\Lambda,0}a_{\min}^2+\Omega_{k,0}a_0^2\right).
$$
Taking the positive root on the expanding branch and eliminating $a_0$ with the result of part (iii),
$$
\boxed{\dot a_{\min}=H_0a_{\min}\left[3\Omega_{\Lambda,0}+\Omega_{k,0}\left(\frac{2\Omega_{\Lambda,0}}{\Omega_{m,0}}\right)^{2/3}\right]^{1/2}.}
$$
The bracket must be positive for a genuine passage through this <minimum expansion speed in a matter-Lambda universe>. Negative values mean that this formal value of the <scale factor> is not accessible on the corresponding physical expanding solution.
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