Let $(A,V,\|\cdot\|_V)$ be a $n$-dimensional normed affine space where $A$ is the point set and $V$ is a $F$-vector space, and $F$ is equipped with a modulus $|\cdot|$. Let $(B,W,\|\cdot\|_W)$ be a one-dimensional normed affine space in which $W$ is a one dimensional $F$-vector space. If we choose an origin $b\in B$ and a non-zero vector ${\bf w}\in W$ then every point in $B$ can be written as $b+c{\bf w}$ for some $c \in F$.
Here all norms and modulus are real-valued. Again, note $B$ and $W$ are ONE-DIMENSIONAL.
1. Directional derivative
A function $f:A \to B$, we say $f$ is differentiable at $a \in A$ along direction ${\bf{v}}\in V$ if there exists some ${\bf u} \in W$ s.t.
$\lim_{h→0}\frac{‖f({ a}+h{\bf v})-f({ a})-h{\bf u}‖_W}{|h|}=0$
where $h \to 0$ is a notation for $|h| \to 0$. We say
$\frac{\partial f({ a})}{\partial {\bf v}}={\bf u}$
is the derivative of $f$ along direction $\bf {v}$ at $ a$.
2. Local maximum and minimum along a direction
When an origin $b\in B$ and a base ${\bf w} \in W$ are chosen, a $f:A \to B$ map is equivalent to a corresponding $f_F:A\to F$ map since every point in $B$ is uniquely mapped to a coordinate in $F$.
A function $f:A\to B$ achieves a local maximum at point $a^* \in A$ along direction ${\bf v} \in V$ if there exists $\epsilon > 0$ s.t. $|f_F(a^*+h{\bf v})|\le|f_F(a^*)|$ for all $|h|<\epsilon$.
Similarly $f$ achieves a local minimum at $a^* \in A$ along direction $\bf v$ if there exists $\epsilon > 0$ s.t. $|f_F(a^*+h{\bf v})|\ge|f_F(a^*)|$ for all $|h|<\epsilon$.
Problem
Now the problem is to prove if $f$ is differentiable along direction $\bf v$ near and at a local maximum or minimum $a^*$, then
$\frac{\partial f({ a^*})}{\partial {\bf v}}={\bf 0}$