calculus of variations: does extremum squared give the same solution?

Question

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.