= Solution
For fixed initial and final coordinates, the two <Euler-Lagrange equations> are
$$
\boxed{\frac d{dt}\left(\frac{\partial f}{\partial\dot x}\right)-\frac{\partial f}{\partial x}=0,\qquad\frac d{dt}\left(\frac{\partial f}{\partial\dot y}\right)-\frac{\partial f}{\partial y}=0.}
$$
Indeed the <first variation> of the <functional> is
$$
\delta F=\int_0^T(f_x\delta x+f_y\delta y+f_{\dot x}\delta\dot x+f_{\dot y}\delta\dot y)\,dt.
$$
<Integration by parts> moves the time <derivatives> from the <variations>; their endpoint values are zero. The <fundamental lemma of the calculus of variations> then gives the two equations above.
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