Let $X$ be a random variable that is Beta distributed with parameters $\alpha=\beta=\theta>0$, so the given frequency function is
$f(x|\theta)=\dfrac{\Gamma(2\theta)}{\Gamma(\theta)^2}x^{\theta-1}(1-x)^{\theta-1}1_{(0,1)}(x)$.
Where $\Gamma(\theta)$ is the gamma function. I want to test $H_0:\theta=1$ against $H_1:\theta=2$. With the LR-test I found that you reject $H_0$ for large values of $X(1-X)$. Now I'm supposed to show that we should reject the null hypothesis for values of $X$ within the interval $[1/2-c,1/2+c]$ for a certain $c$ (constant) that is later determined. Can someone help me out with this?