I have a question regarding this fact:
"Recall that a concave function $u:X \rightarrow \mathbb{R}$ attains its maximum at a point $x$ if and only if its directional derivative $u'(x;d)$ at point $x$ in any direction $d \in X-\{x\}$ is non-positive."
where $X$ is a non-empty, compact and convex subset of $\mathbb{R}^n$ (for some $n \in \mathbb{N}^\ast$).
For simplicity, consider the case where $n = 1$ and $u: [a,b] \subset \mathbb{R} \rightarrow \mathbb{R}$ is affine and strictly increasing ($a,b \in \mathbb{R}$, $0 < a < b$).
First question: Can we talk about directional derivative at points $a$ and $b$ (I think that this notion is defined only when the domain of $u$ is open) ? I guess that we have two options: $(1)$ take the right directional derivative for $a$ and the left directional derivative for $b$, or $(2)$ extend $u$ to another function $U: O \subset \mathbb{R} \rightarrow \mathbb{R}, x \mapsto u(x)$, where $O$ is open and define the directional derivatives of $a$ and $b$ using $U$. In general, do we use one of this option or something different ?
Second question: In that case, $u$ attains its maximum at $b$, but considering the two previous options: $(1)$ I don't think that the directional derivative $u'(b,d)$ at point $b$ in any direction $d \in [a,b[$ is non-positive and $(2)$ I don't even have the feeling that it is a good idea to proceed like this, because I think that the directional derivatives at points $a$ and $b$ will depend on $U$... Could you tell me where I'm wrong/missing something ?
Thank you for you help !
The (one sided) directional derivative is defined as (notations differ) $u'(x,d) = \lim_{t \downarrow 0} {u(x+td)-u(x) \over t}$.
It is straightforward to show (see here for some discussion) that for a concave function $f$ defined on an interval the quantity $R(t_1,t_2) = {f(t_1)-f(t_2) \over t_1-t_2}$ (with $t_1 \neq t_2$, of course) is symmetric in $t_1,t_2$ and non increasing in $t_1$ for a fixed $t_2$.
In particular, $u'(x,d) = \sup_{t >0} {u(x+td)-u(x) \over t}$ is well defined (it may be $+\infty$) for any direction $d \in X \setminus \{x\}$.
In your example, $u$ is affine so $u(x) = v+tg$ for some $x,w$ and the directional derivative is given by