Summary: This paper investigates the minimization problem in \(n\)-dimensional Euclidean spaces, and analyzes some basic properties of the Lagrangian dual of the problem. Two new results on the Lagrangian dual function are further obtained. We first prove that there exists an element in an arbitrary nonempty set which can be utilized to formulate the directional derivative of the Lagrangian dual function at any given point. And then we obtain a simpler and more straightforward conclusion that there only exists the same type of subgradients compared with that in a classic result. The mathematical expression of the directional derivative of the Lagrangian function is thus simplified.