# Minimizing an integral with variable endpoints

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?

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Consider $T$ fixed. Then you have a standard variational problem and you can use the standard conditions to check whether the stationary solution is a minimum. That gives you a function $J(T)$ of one variable $T$, and again you can use the standard conditions to check whether its stationary point is a minimum. If $J(T)$ is the minimum of $J_T[y]$ and $J(2)$ is the minimum of $J(T)$, then $J(2)$ is the minimum of $J_T[y]$.