1. **Problem Statement:** Minimize the objective function subject to given constraints. Since the specific function and constraints are not provided, let's outline the general approach. 2. **General Approach:** - The objective function is usually denoted as $f(x)$ or $f(x,y,\ldots)$. - Constraints can be equalities $g_i(x) = 0$ or inequalities $h_j(x) \leq 0$. 3. **Method:** - For problems with constraints, use methods like Lagrange multipliers or KKT conditions. - For unconstrained problems, set the gradient $\nabla f = 0$ and solve. 4. **Lagrange Multipliers:** - Form the Lagrangian $$L(x, \lambda) = f(x) + \sum_i \lambda_i g_i(x)$$ - Find stationary points by solving $$\nabla_x L = 0$$ and $$g_i(x) = 0$$. 5. **Check Constraints:** - Verify that solutions satisfy all constraints. - Check second-order conditions or use Hessian to confirm minima. 6. **Summary:** - Without the explicit function and constraints, this is the general framework. - Provide the function and constraints for a detailed solution.