A question from my Analysis list that I could not have any idea. Any help would be great. I don't want a complete solution, just a little hint, because I need to do at least one part alone.
Let $f: U \longrightarrow \mathbb{R}$ differentiable in open $U \subset \mathbb{R}^{m}$. Suppose that $df(a) \neq 0$ for $a \in U$ and unitary vector $u \in \mathbb{R}^{m}$ such that $df(a)u = \max \lbrace df(a)h\,|\,|h|=1 \rbrace$. If $v \in \mathbb{R}^{m}$ is such that $df(a)v=0$, show that $v$ is perpendicular to $u$.
Small hints
The question has nothing to do with real anlysis: if $A \neq 0$ is a $1\times m$ matrix and $u$ such that: $$A u =\max_{|h|=1} A h$$ and if $A v=0$ then $\langle u,v \rangle =0$.
(What can be said of $A\frac{u+t v}{|u+tv|} $?)