 Solution

With fixed endpoint values, the Euler-Lagrange equation is $F_y-\frac{d}{dx}{F}_{y^{\prime }}=0$. If $F$ has no explicit $x$ dependence, differentiation givesThus the Beltrami identity is $F-y^{\prime }{F}_{y^{\prime }}=\text{constant}$.

For this integrand, $F_y=2y$ and $F_{y^{\prime }}=1+2y^{\prime }$, so $y^{\prime \prime }=y$. The extremals arewhere the two fixed endpoint values determine $A,B$. The Beltrami constant is $y^2-y^{\prime 2}$, consistent with this solution.

For a nonzero smooth perturbation $\eta$ vanishing at the endpoints, expand the quadratic functional. The first variation vanishes at an extremal, whileThe linear term $\int y^{\prime }$ depends only on the fixed endpoints. Consequently the extremal is a strict minimum and cannot be a maximum.