The exercise
We have $\Theta = \{0,1\}$ and let $X$ be a random variable with density function $f(x;0)=1$ and $f(x;1)=3x^2$ for $x\in (0,1)$. I want to find the most powerful test of size $\alpha=0.2$ for the testing problem: $H_0: \theta=0$ vs $H_1: \theta=1.$
I have done a similar exercise for a given discrete distribution, but this one is quite different.
What I have done so far
My first thoughts were to let $X_1,...,X_n$ be a random sample from $X$, and then search for a critical region $$R=\left\{\underset{\sim}{X}: \frac{f_1(\underset{\sim}{X})}{f_0(\underset{\sim}{X})} > k\right\} ,$$ and then choose $k$ such that $P_{\theta_0}(\underset{\sim}{X} \in R)=\alpha.$
The ratio can then be calculated as
$$\frac{f_1(\underset{\sim}{X})}{f_0(\underset{\sim}{X})} = \frac{f_1(\underset{\sim}{X};1)}{f_0(\underset{\sim}{X};0)} = 3 \sum_{i=1}^n X_i^2,$$
and therefore
$$P_{\theta_0}\left\{ \frac{f_1(\underset{\sim}{X})}{f_0(\underset{\sim}{X})}>k\right\} =P_{\theta_0}\left\{3 \sum_{i=1}^n X_i^2 >k\right\}.$$
This is where I become stuck, so I would really appreciate some feedback on a) if I have done the right stuff so far, and b) How I should now proceed.
