The original version was totally incorrect. This question can be removed. Sorry for the bad post.
2026-03-27 00:54:57.1774572897
Euler-Lagrange example proof
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Your approach doesn't work because your Euler-Lagrange equation is inconsistent. Since $$J=3y\left( 2\right) -3y\left( 1\right) - \frac{2^2-1^2}{2} + 2\int_1^2 ydx=\frac{9}{2} + 2\int_1^2 ydx,$$the task is equivalent to extremising $\int_1^2 ydx$. That the Euler-Lagrange equation is still inconsistent hints at the fact that there is no extremal choice of $y$. The boundary values are consistent with $y=2x+1+A\left( x-1\right)\left( x-2\right)$, so we can give $J$ any value with suitable $A$.