= Solution
For a vector equality constraint, take the <Lagrangian function in constrained optimization>
$$
\boxed{L(x,\lambda)=f(x)+\lambda^T(b-g(x)),\qquad x\in X,\quad\lambda\in\mathbb R^m.}
$$
The sign of the multiplier is a convention; equality multipliers are unrestricted. The <Lagrange sufficiency theorem> says that if $x^*\in X$ is feasible and, for some $\lambda^*$, globally minimizes $L(\cdot,\lambda^*)$ over $X$, then $x^*$ globally minimizes the constrained objective.
Indeed for any feasible $x$, $f(x)=L(x,\lambda^*)\ge L(x^*,\lambda^*)=f(x^*)$. Thus $\boxed{x^*\text{ is a global primal minimizer}}$. The theorem requires a global Lagrangian minimum, not merely a stationary point. It needs neither differentiability nor convexity when stated in this form; those hypotheses can be used separately to establish the required minimum.
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