In this question the solution of Euler–Lagrange equation is $y=x$ function.
$L = (y')^3$ so $L''_{y'} = 6y'$ and is positive when $y=x$. But from the answer of Emanuele Paolini follows that it is not enough to $y$ be a minimum.
So what is the general method of checking whether a stationary point is extremum?
If everything with your example is ok, then the function y(x) = x will be a local minimizer. If your functional is not convex (as in your case) it is very hard to verify that you have found a global minimizer.
Maybe the best chance is the so-called zero-th order sufficient condition: The objective $J$ satisfies $J(f) \ge j$ for some $j \in \mathbb{R}$ and all feasible $f$ and $J(\bar f) = j$ for some feasible $\bar f$.