This paper presents an analytical solution to a constrained optimization problem with two variables,x1 and x2, where the objective function is as follows: \( f\left(x1,x2\kern0em \right)={x}_1^2+{x}_2^2 \) . The problem is subject to an equality constraint h(x1, x2) = (1 − x1)(1 − x2) = 0 and box constraints0 ≤ x1, x2 ≤ 1. Using the Karush-Kuhn-Tucker (KKT) conditions, we derive two potential solutions:(1, 0) and (0, 1). The sufficiency of these solutions is verified through an eigenvalue analysis of the Hessian matrix, revealing that while the solutions satisfy all KKT conditions and are local minima, the boundary constraints prevent them from being strict minima. The results emphasize the role of boundary constraints in shaping the optimality of the solution and highlight the application of both necessary and sufficient conditions in constrained optimization problems. This distinction is crucial when interpreting the solutions in constrained optimization problems domain, where the boundary of the feasible set can affect the strictness of the minima points. This study signifies the value of considering both the KKT conditions and second-order optimality criterion, such as the eigenvalue analysis of the Hessian, when constrained optimization problems are analytically solved. Furthermore, it shows the interpretation of solutions when boundary constraints affect the optimization landscape, leading to the classifying the solutions as local minima rather than strict minima.