I was working on the following problem:
Consider two probability density functions on $[0,1]: f_0(x) = 1$, and $f_1(x) = 2x$. Among all tests of the null hypothesis $H_0: X \sim f_0(x)$ versus the alternative $X \sim f_1(x)$, with significance level $\alpha = 0.1$, how large can the power possibly be?
I think we need to begin by looking at an arbitrary test with significance level $\alpha = 0.1$ but I am having a hard time doing this. I am not even sure this is the direction we want to head in so I was hoping to get some hints regarding this problem.