= Solution
For $0<\pi<1$, let $\delta_\pi$ be the <Bayes classifier> with class-zero prior $\pi$, and define its two class-conditional error probabilities by
$$
e_0(\pi)=\mathbb P_0(\delta_\pi(X)=1),
\qquad
e_1(\pi)=\mathbb P_1(\delta_\pi(X)=0).
$$
Writing the <likelihood ratio> as $L=f_1/f_0$, the rule chooses class one when
$$
L(x)\geq\frac\pi{1-\pi}.
$$
The Gaussian likelihood-ratio level sets have probability zero, so <dominated convergence theorem>[dominated convergence] shows that $e_0$ and $e_1$ depend continuously on $\pi$. As $\pi\downarrow0$, the threshold tends to zero and the classifier chooses class one almost surely, giving
$$
(e_0(\pi),e_1(\pi))\longrightarrow(1,0).
$$
As $\pi\uparrow1$, it chooses class zero almost surely, giving the opposite limit $(0,1)$. The <intermediate value theorem> therefore provides $\pi^*\in(0,1)$ such that
$$
e_0(\pi^*)=e_1(\pi^*)=:r.
$$
The rule $\delta_{\pi^*}$ is an <equalizer rule> with worst-case risk $r$. For any classifier $\delta$,
$$
\begin{aligned}
\max\{R_0(\delta),R_1(\delta)\}
&\geq \pi^*R_0(\delta)+(1-\pi^*)R_1(\delta)\\
&\geq \pi^*e_0(\pi^*)+(1-\pi^*)e_1(\pi^*)=r,
\end{aligned}
$$
because $\delta_{\pi^*}$ minimizes the integrated risk for its prior. Hence $\delta_{\pi^*}$ is a <minimax decision rule>, as summarized by the <minimax Gaussian Bayes classifier from equal class errors>.
The prior is indeed <least favorable prior>[least favorable]. Its <Bayes risk> is $r$, while for any other prior $q$ the Bayes risk is at most the integrated risk of the same equalizer rule $\delta_{\pi^*}$, namely
$$
q r+(1-q)r=r.
$$
\b[Thus no prior has larger Bayes risk.]
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