Editing Calculus of variations (section)

== Du Bois-Reymond's theorem ==

The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral <math>J</math> requires only first derivatives of trial functions. The condition that the first variation vanishes at an extremal may be regarded as a '''weak form''' of the Euler–Lagrange equation. The theorem of Du Bois-Reymond asserts that this weak form implies the strong form. If <math>L</math> has continuous first and second derivatives with respect to all of its arguments, and if
<math display="block">\frac{\partial^2 L}{\partial f'^2} \ne 0,</math>
then <math>f</math> has two continuous derivatives, and it satisfies the Euler–Lagrange equation.