Critical value of the most powerful test for the rate of a gamma density

Let $X$ be an observation from a population with density

$$f(x)=\begin{cases}\lambda^2xe^{-\lambda x}, & x>0, \lambda>0,\ 0, & \text{elsewhere.}\end{cases}$$

For testing $H_0:\lambda=2$ against $H_1:\lambda=1$, the most powerful test of size $\alpha$ rejects $H_0$ if $X>c$, where $c$ is given by