Let $p \in \mathbb{N}^{\star}$. We assume that $f, g_1, \ldots, g_p$ are functions from $\mathbb{R}^n$ to $\mathbb{R}$ differentiable on $\mathbb{R}^n$, and that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. For all $x \in K$, we denote $I_x = \left\{ i \in \llbracket 1, p \rrbracket, g_i(x) = 0 \right\}$. Show that for all $x \in K$, $$\mathcal{A}_K(x) \subset \left\{ h \in \mathbb{R}^n, \forall i \in I_x, \langle \nabla g_i(x), h \rangle \leqslant 0 \right\}.$$
Let $p \in \mathbb{N}^{\star}$. We assume that $f, g_1, \ldots, g_p$ are functions from $\mathbb{R}^n$ to $\mathbb{R}$ differentiable on $\mathbb{R}^n$, and that
$$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$
is non-empty. For all $x \in K$, we denote $I_x = \left\{ i \in \llbracket 1, p \rrbracket, g_i(x) = 0 \right\}$.\\
Show that for all $x \in K$,
$$\mathcal{A}_K(x) \subset \left\{ h \in \mathbb{R}^n, \forall i \in I_x, \langle \nabla g_i(x), h \rangle \leqslant 0 \right\}.$$