= Solution
For fixed endpoint values and smooth admissible variations, the <Euler-Lagrange equation> is
$$
\boxed{\frac{d}{dx}\frac{\partial f}{\partial z'}-\frac{\partial f}{\partial z}=0.}
$$
The first variation follows by integrating $\int(f_z\eta+f_{z'}\eta')dx$ by parts and using $\eta=0$ at the endpoints.
For the claimed first <integral>, differentiate directly:
$$
\frac{d}{dx}(f-z'f_{z'})=f_x+z'\left(f_z-\frac{d}{dx}f_{z'}\right).
$$
Along an extremal the bracket vanishes. Thus when $f_x=0$, the <Beltrami identity> is
$$
\boxed{f-z'f_{z'}=\text{constant}.}
$$
For a further regular constraint $G[z]=G_0$, introduce a constant <Lagrange multiplier> $\lambda$ and make $F-\lambda G$ stationary. Apply the <Euler-Lagrange equation> to the combined integrand, then impose the endpoints and $G=G_0$ to determine the <integration> constants and multiplier.
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