I am trying to minimize the following functional: $ J[y]=\int\limits_0^T{\frac{\sqrt{1+y'(x)^2}}{y(x)}dx}, $
$ y(0)=1, ~ T-y(T)=1, $
where $T$ is variable.
Using the necessary conditions I've found that
$ (y(x),~T) = (\sqrt{(2-(x-1)^2)},~2) $
is the extremum. But I can't prove that this extremum is actually minimum. Are there any theorems used to check if sufficient conditions are satisfied for these kinds of problems?