# Variational calculus … Prove that a linear functional $\varphi [h]$ cannot have an extremum unless $\varphi [h] \equiv 0$

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.

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