<![CDATA[This study develops a mathematical optimization model for determining the optimal debt ratio in a firm’s capital structure using the Lagrange method. By expressing equity value as a function of leverage, the model identifies a structural asymptote that divides the feasible region into positive and negative equity outcomes, thereby eliminating the possibility of an interior maximum and mathematically refuting the central prediction of the Trade-Off Theory. The first-order condition produces two critical points, but only one lies within the positive region and is shown through second-order analysis to be a local minimum rather than an optimum. Numerical simulation and empirical validation further demonstrate that optimal equity value arises under two extreme leverage conditions: zero leverage for conservative strategies and high leverage approaching the asymptote for aggressive strategies. These results establish a new analytical foundation for capital structure theory, provide a basis for developing more complex econometric models, and offer practical guidance for firms in prioritizing leverage decisions according to their risk preferences and strategic objectives]]>
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