Let $\mathcal{D} \subset \mathbb{R}^n$ be a convex set. Let $f: \mathcal{D} \to \mathbb{R}$ be a differentiable function. Show that the following are equivalent:
(a) $f$ is concave on $D$. (b) $f(y) -f(x) \le D f(x) (y-x)$ for all $x, y \in \mathcal{D}$. (c) $[Df(y) - Df(x)] (y-x) \le 0$ for all $x, y \in \mathcal{D}$.
The textbook provides the proof for (a) $\iff$ (b). For (b) $\implies$ (c), (b) also implies that $D f(y) (x -y) \ge f(x) - f(y)$. Combining two results, we can get (c). So, I am left with one implication. I need to show either (c)$\implies$ (a) or (c) $\implies$ (b). I don't know how to proceed. I would appreciate if you give some hint.
