 Affine Euler-Lagrange solution from finite derivative fibers

For an autonomous scalar integral $\int f(u^{\prime })$, the first variation gives the integrated Euler-Lagrange equation $f^{\prime }(u^{\prime })=c$. If $u$ is $C^1$ and each fiber of $f^{\prime }$ is finite, the connected image of the interval under the continuous derivative $u^{\prime }$ lies in a finite set and is therefore a singleton. The fixed endpoints then determine exactly one affine solution. If $f$ is strictly convex, the strict Jensen's inequality applied to $u^{\prime }$ additionally makes this solution the unique global minimizer among fixed-endpoint Lipschitz functions. Without convexity the same unique smooth stationary solution can be a maximizer, for example at zero for $f(v)=1/(1+v^{2008})$ and zero endpoint values.