Defining carefully the terminology that you use, state and prove the NeymanPearson Lemma.

Let $X$ be a single observation from the distribution with density function

$$f(x \mid \theta)=\frac{1}{2} e^{-|x-\theta|}, \quad-\infty<x<\infty$$

for an unknown real parameter $\theta$. Find the best test of size $\alpha, 0<\alpha<1$, of the hypothesis $H_{0}: \theta=\theta_{0}$ against $H_{1}: \theta=\theta_{1}$, where $\theta_{1}>\theta_{0}$.

When $\alpha=0.05$, for which values of $\theta_{0}$ and $\theta_{1}$ will the power of the best test be at least $0.95$ ?