Solution
= Solution
Differentiate
$$
Q=f-\left(f_{y'}-\frac d{dx}f_{y''}\right)y'-f_{y''}y''.
$$
The product rule cancels all terms except
$$
Q'=y'\left(f_y-\frac d{dx}f_{y'}+\frac{d^2}{dx^2}f_{y''}\right)+f_x.
$$
On an extremal with no explicit $x$ dependence both terms vanish, proving the stated conserved quantity.