Type I and Type II errors of a threshold test for an exponential rate

Let $X$ be a single observation from a population having an exponential distribution with mean $1/\lambda$. Consider the problem of testing $H_0:\lambda=2$ against $H_1:\lambda=4$. For the test with rejection region $X\ge3$, let $\alpha=P(\text{Type I error})$ and $\beta=P(\text{Type II error})$. Then