Do the following two integrals have the same solution? $J=\int^{{x}_2}_{{x}_1}f(y, \dot{y}, x)dx$, $J=\int^{{x}_2}_{{x}_1}f(y, \dot{y}, x)^2dx$. Several questions, such as shortest line in a plane, shortest line on a sphere, minimum surface of revolution, have the same solution using the two integrals. But I cannot prove it. Any help would be appreciated.
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It depends on $f$. If function does not take negative values, extrema will be achieved at the same points for $f^2$ because $x^2$ is strictly increasing function, i.e. $x_1>x_2 \implies x_1^2>x_2^2$ and vise versa – Vasili Jun 30 '22 at 12:34