Calculus of variations Euler-Lagrange equation and variational problem 
Find all the extrema (local minima and maxima) of the function $$J[y] = \int\limits_1^2(xy' + y)^2\,\mathrm dx;\qquad y(1) = 1, y(2) = \dfrac12.$$
Hint. Once you've found the solution of the Euler-Lagrange equation with the boundary conditions, remember to check, like in the previous problem, if this solution is a minimum, a maximum or not an extremum.


The image above shows my work. I'm pretty sure I solved the E-L equation correctly with the boundary conditions, but I am not too sure about the variation part. I always seem to find an absolute minimum, which makes me think my understanding of this part is lacking.

Alternatively, one can change the variable $$z(x)~:=~xy(x).$$
Then OP's variational problem simplifies to
$$I[z] ~:=~ \int_1^2z^{\prime 2}\,\mathrm dx~\geq~0;\qquad z(1)~=~1~=~z(2).$$
Note that the functional $I$ is non-negative.
The Euler-Lagrange (EL) equation becomes $z^{\prime\prime}=0$. With the correct BCs, the EL equation leads to the constant solution $z_{\ast}(x)=1$. Since $I[z_{\ast}]=0$, the unique stationary solution $z_{\ast}$ minimizes the functional $I$. 
