Show that the likelihood ratio test of 395 still holds

Question

In this problem we generalize the likelihood ratio test for simple binary hypotheses by including costs incurred in the decision-making process.

Let C_ij denote the cost incurred in deciding in favor of hypothesis H_i when hypothesis H_j is true.

Hence, show that the likelihood ratio test of (3.95) still holds, except for the fact that the threshold of the test is now defined by