A single observation obtained from Gumbel distribution has pdf $f(x)$ = $e^{-(x-a) - e^{-(x-a)}}$.
Show that most powerful test for $H_0$: $a=0$ vs. $H_A$ : $a= a_1$ where $a_1 > 0$ has rejection region RR = ${x>c}$ where $c$ is constant.
I don't understand what question asking. I did likelihood test and $$\frac{L(\theta_o)}{L(\theta_a)} = \frac {e^{-x-e^{-x}}} {e^{-(x-{a_1})-e^{-(x-{a_1})}}}$$ = $$ e^{(e^a - 1)(e^{-x})-a} $$
But there is no range for x or alpha given, so what is next step?