= Solution
Apply the <minimum-integral decision region> principle to $g(x)=\pi_2f_2(x)-\pi_1f_1(x)$. The optimal <Bayes classifier> chooses class one when $\pi_1f_1(x)>\pi_2f_2(x)$, and class two for the reverse inequality. Ties can be assigned either way. An everywhere valid discriminant score is
$$
\boxed{\delta(x)=\pi_1f_1(x)-\pi_2f_2(x),\qquad\text{choose class one if }\delta(x)>0.}
$$
Where both weighted densities are positive one may equivalently use $\log(\pi_1f_1(x)/(\pi_2f_2(x)))$ and threshold zero. This chooses the largest posterior class probability, because both posterior probabilities have the same positive mixture-density denominator.
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