DSE 2012 Question 39: Hypothesis testing

Suppose you have a sample of size one from one of the following densities $H_{0}:\ f(x)=2x,\ 0\le x\le 1$ and $H_{1}:\ f(x)=2-2x,\ 0\le x\le 1$. Let $\alpha$ and $\beta$ denote type I error and type II error, respectively. Find the test procedure of the form 'Reject $H_{0}$ if $x<k$' with $\alpha=0.09$. Find $\beta$ for this test.