Neyman–Pearson best critical region

Let $x_1,x_2,\dots,x_n$ be a random sample from the distribution having p.d.f. $f(x;\theta,\beta)=\beta e^{-\beta(x-\theta)}; x\ge\theta,\ \beta>0$. To test $H_0:\beta=2,\theta=3$ against $H_1:\beta=4,\theta=2$, the best critical region is given by:

where $k>0$.