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 $I_x = \left\{ i \in \llbracket 1, p \rrbracket, g_i(x) = 0 \right\}$. Suppose that $f$ attains at $x^{\star} \in K$ a local minimum on $K$, and that hypothesis $(H)$ is verified. Show that there exist non-negative real numbers $\mu_1^{\star}, \ldots, \mu_p^{\star}$ such that $$\left\{ \begin{array}{l} \nabla f(x^{\star}) + \sum_{i=1}^{p} \mu_i^{\star} \nabla g_i(x^{\star}) = 0 \\ \mu_i^{\star} g_i(x^{\star}) = 0 \text{ for all } i \in \llbracket 1, p \rrbracket. \end{array} \right.$$
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 $I_x = \left\{ i \in \llbracket 1, p \rrbracket, g_i(x) = 0 \right\}$.\\
Suppose that $f$ attains at $x^{\star} \in K$ a local minimum on $K$, and that hypothesis $(H)$ is verified. Show that there exist non-negative real numbers $\mu_1^{\star}, \ldots, \mu_p^{\star}$ such that
$$\left\{ \begin{array}{l} \nabla f(x^{\star}) + \sum_{i=1}^{p} \mu_i^{\star} \nabla g_i(x^{\star}) = 0 \\ \mu_i^{\star} g_i(x^{\star}) = 0 \text{ for all } i \in \llbracket 1, p \rrbracket. \end{array} \right.$$