= Solution
The <Neyman-Pearson lemma> says that for simple hypotheses with densities $f_0,f_1$, a size-$\alpha$ test that rejects where $f_1/f_0>k$ and accepts where $f_1/f_0<k$, with possible randomization on equality to attain size $\alpha$, is most powerful among all tests of size at most $\alpha$. To prove it, let $\varphi^*$ be this test and $\varphi$ any competing test. Pointwise,
$$
(\varphi^*-\varphi)(f_1-kf_0)\geq0.
$$
After integration,
$$
\mathbb E_1\varphi^*-\mathbb E_1\varphi
\geq k(\mathbb E_0\varphi^*-\mathbb E_0\varphi)\geq0,
$$
which proves maximal power.
For the exponential sample, with $S=\sum_iX_i=n\overline X$,
$$
\frac{L(\lambda_1)}{L(\lambda_0)}
=\left(\frac{\lambda_1}{\lambda_0}\right)^n
\exp\bigl((\lambda_0-\lambda_1)S\bigr).
$$
Since $\lambda_1<\lambda_0$, this ratio is increasing in $S$. The <likelihood-ratio test> therefore rejects for large $\overline X$:
$$
\boxed{R_\alpha=\{\overline X>c_\alpha\},
\qquad
1-G_{n,\lambda_0}(nc_\alpha)=\alpha},
$$
where $G_{n,\lambda}$ is the distribution function of the rate-parameterized <Gamma distribution> $\Gamma(n,\lambda)$.
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