Consider a minimization problem $(P)$ :
minimize $f(x)$
subject to $\delta_C(x) \leq 0$
Now assume that $\emptyset \neq C \subset \mathbb{R}^n$ is convex and let $f: \mathbb{R}^n \to \mathbb{R}$ be convex and differentiable.
How many I prove that $\bar x \in C$ is a solution of $(P)$ iff $\langle \nabla f(\bar x), y - \bar x \rangle \geq 0, \space \forall \space y \in C $
Additionally, if $C$ is open, then $\bar x$ is a solution of $(P)$ iff $\nabla f(\bar x) =0 $
Can I clarify that $\langle \nabla f(\bar x), y - \bar x \rangle$ is the directional derivative at $\bar x$ in the direction of $y-\bar x $ ?
A part from that, geometrically what happens if the gradient vector is negative?
For the additional statement, why does the gradient vector have to be $0$. What is the geometric interpretation of this?