= Solution
\b[The approximate population peaks about $11.3\ \mathrm h$ after flux removal.] Differentiate the preceding difference of exponentials. The stationary condition is
$$
-\lambda_Ie^{-\lambda_It}+\lambda_Xe^{-\lambda_Xt}=0,
$$
so for distinct <nuclear decay constants>,
$$
\boxed{
t_{\max}=\frac{\ln(\lambda_I/\lambda_X)}{\lambda_I-\lambda_X}.
}
$$
Using hours consistently,
$$
\lambda_I=\frac{\ln2}{6.7}=0.103455\ \mathrm{h^{-1}},
\qquad
\lambda_X=\frac{\ln2}{9.2}=0.0753421\ \mathrm{h^{-1}},
$$
and therefore
$$
\boxed{t_{\max}=11.2794\ \mathrm h\simeq11.3\ \mathrm h.}
$$
The approximate population is positive for $t>0$, starts at zero, tends back to zero and has only this one stationary point, so it is a maximum.
\Image[/past-exam-of-the-mphil-in-nuclear-energy-course-of-the-university-of-cambridge/2019/paper-ne1-precursor-response.png]
{title=Normalized precursor-fed absorber response after source removal}
The delayed maximum corresponds to a delayed maximum in the magnitude of the absorber's negative contribution to <reactor reactivity>, within the fixed model. It does not determine a permissible restart time, which would require information absent here. The estimate also inherits the high-flux initial-population approximation of part (ii); the exact solution can peak at a different time when that approximation is unsuitable.
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