Find the Neyman-Pearson test with size $\alpha$ to contrast $$ H_0: \beta = 1$$ $$ H_1: \beta = \beta_1$$ with $\beta_1$ > 1 based in a sample of size 1 of the random variable with density: $$f(x,\beta) = \beta x^{\beta-1}, \ \ 0<x<1 $$
I've found that the rejection region is
$$C = \{ x | \ \beta_1 x^{\beta_1-1} > k \} $$ I've been told that it's similar to $$C´ = \{ x | x < k´ \} $$ because C seems to decrease as x increase, but I don't understand why. Could you explain me why these two expressions are the same to solve the problem?
The critical region you have found be NP Lemma,
$$\frac{f_1(x)}{f_0(x)} \ge k$$ $$\implies x^{\beta_1-1} \ge k', \quad \beta_1 > 1$$ $$\implies x \ge k''$$
So we have that we'll reject $H_0$ if $x \ge k''$.
The second critical region you have shown is not correct.