From definitions in the book calculus of variations - Gelfand and fomin http://web.cs.iastate.edu/~cs577/handouts/variations.pdf
I try prove that a linear functional $\varphi[h]$ cannot have an extremum unless $\varphi[h] \equiv 0$.
I tried the following 1- prove that $\varphi[h]$ is differentiable ann use that theorem 2 pag 13 in the book
2- I tried the use that a $J[y]$ has an extremum in $t$ if $J[y] -J[t]$ does not change its sing in some neighborhood, but for me it is not clear the not change of sign of $\varphi[h]$
Any help is good. Thanks.