This is an exercise in Cartan's Differential Forms:
If $f(t,x,y)$ satisfies $$\frac{\partial{f}}{\partial x}-\frac{\partial^2{f}}{\partial t\partial y}-y\frac{\partial^2{f}}{\partial x\partial y}=0$$ and $f$ is convex as a function of $y$, then $$\int_a^bf(t,x,x')dt\ge\int_a^bf(t,X,X')dt$$ where $x(t)$ and $X$(t) has the same endpoint and $X$ is straight line as a graph.
It is easy to see $X$ is an extremum directly from the Euler-Lagrange equation, but I have no clue how to show $X$ is minimal from that convex condition.