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(Solved): Prove that one and only one extremal of each of the functionals \int e^(-2y^(2))(y^('2)-1)dx,\int (y ...
Prove that one and only one extremal of each of the functionals
\int e^(-2y^(2))(y^('2)-1)dx,\int (y^(2)+y^(')tan^(-1)y^(')-ln\sqrt(1+y^('2)))dx
passes through any two points of the plane with different abscissas.
Hint. Apply Bernstein's theorem.