Page 52, Coleman, Rodney. Calculus on normed vector spaces. Springer Science & Business Media, 2012.
It is a corollary about extremum on normed vector space. I got lost about this proof. It assumed $f'(x) v >0$ firstly then got a contradiction $f'(x) v <0$, but I didn't find it uses the $f'(x) v>0$ assumption during this process. Any help would be appreciated!
Pick $v$ and define $\phi(t) = f(x+tv)$. Since $x$ is a local extremum we have $\phi'(0) = 0$. It is straightforward to check that $\phi'(0) = f'(x) v$.