= Solution
Starting with the <electron neutrino> state in <3/a>, the survival <probability amplitude> is
$$
S_{ee}(t)=c^2e^{-i\chi_1t}+s^2e^{-i\chi_2t}.
$$
Taking its squared <absolute value> gives
$$
P_{ee}=1-\sin^22\theta\sin^2\frac{\delta\chi t}{2}
=\cos^22\theta+\sin^22\theta\cos^2\frac{\delta\chi t}{2}.
$$
This derives the supplied survival <probability> from the evolution operator rather than from a Hamiltonian entry. It also shows why the two written forms agree. For the stipulated phase and $\theta=22.5^\circ$, $\sin^22\theta=1/2$, so at $L=79\ \mathrm{km}$,
$$
\boxed{P_{ee}(E)=\frac12+\frac12\cos^2(7.9/E),\qquad E\text{ in MeV}.}
$$
The phase decreases monotonically from $7.9$ to $0.79$ over the open interval $1<E<10$. Differentiating gives $dP_{ee}/dE=[7.9/(2E^2)]\sin(15.8/E)$. The interior maxima occur at $7.9/E=n\pi$ and the interior minima at $7.9/E=(n+1/2)\pi$. Enumerating those phases gives
$$
\begin{array}{c|c|c}
\text{type}&E\ (\mathrm{MeV})&P_{ee}\\\hline
\text{minimum}&7.9/(5\pi/2)=1.0059&1/2\\
\text{maximum}&7.9/(2\pi)=1.2573&1\\
\text{minimum}&7.9/(3\pi/2)=1.6764&1/2\\
\text{maximum}&7.9/\pi=2.5146&1\\
\text{minimum}&7.9/(\pi/2)=5.0293&1/2
\end{array}
$$
The minimum just above $1\ \mathrm{MeV}$ must not be lost by rounding the lower endpoint. Neither excluded endpoint is an additional extremum.
\Image[/past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2021-neutrino-survival.png]
{title=Reactor survival extrema and the unresolved solar vacuum-oscillation envelope}
For <solar neutrinos> propagating in the same \i[vacuum] model, the vastly longer baseline makes the oscillations extremely fine in <energy>, with the same bounds $1/2$ and $1$. The figure uses a deliberately schematic enlargement of the oscillation spacing to show those bounds. A $1\ \mathrm{MeV}$ detector <energy resolution> averages very many periods, so $\langle\cos^2\varphi\rangle=1/2$ and
$$
\boxed{\langle P_{ee}\rangle_{\rm vacuum}=1-\tfrac12\sin^22\theta=\tfrac34.}
$$
The detector averages event rates weighted by source spectrum, interaction <cross section> and response; the displayed constant follows when these vary slowly across each tiny oscillation period. Reactor production physically gives <electron antineutrinos>. In this two-flavor vacuum approximation their survival <probability> is identical; the positive matter-potential discussion in <3/d> concerns <solar neutrinos>.
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