Extremals of $\int_1^2 (y')^2x^{-\alpha}dx$

For $\alpha\ge0$, consider the functional

$$J_\alpha[y]=\int_1^2\frac{(y')^2}{x^\alpha}\,dx$$

defined for all continuously differentiable functions defined on the interval $[1,2]$ satisfying the conditions $y(1)=1$, $y(2)=2$. Then which of the following statements are true?