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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.

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    $\begingroup$ Use $f(ax) = af(x)$ for suitable choices of $a$ to find points close to $x$ with a larger/smaller value. $\endgroup$ – Winther Jun 8 at 17:12

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