Consider again the conditions of Exercise 4. Show that the p.d.f. f (x|) has a monotone likelihood

Question:

Consider again the conditions of Exercise 4. Show that the p.d.f. f (x|θ) has a monotone likelihood ratio in the statistic r(X) = −X, and determine a UMP test of the following hypotheses at the level of significance α0 = 0.05: H0: θ ≤ 1/2, H1: θ > 1/2 .