12-41. Suppose {X1, . . . ,Xn} is a set of sample random variables drawn from the probability density function f (x; θ) = θxθ−1, 0 < x < 1; zero elsewhere.

Find the most powerful test of size α of H0 : θ = θ0 = 1, against H1 : θ =

θ1 = 2.