The authors examine the types of functionals which can attain their extrema on the solutions of the Euler-Lagrange differential equation by generalizing in a plausible way a result due to Hilbert. Hilbert proved that if we extend a one-dimensional regular functional defined on twice continuously differentiable functions in such a way that we admit once continuously differentiable functions as well, then this extension cannot have any new extremal function. Based on this result the authors consider that the domain of the functional is defined as the largest class of functions where the admissible continuous curves are supposed to have both left and right hand tangents at each point and they choose the simplest one-dimensional non-parametric functionals.