1. **Stating the problem:** We are asked to solve an optimization problem using Lagrangian functions. This typically involves maximizing or minimizing a function $f(x,y,\ldots)$ subject to a constraint $g(x,y,\ldots) = 0$. 2. **Formula and method:** The Lagrangian function is defined as $$\mathcal{L}(x,y,\lambda) = f(x,y) - \lambda (g(x,y) - c)$$ where $\lambda$ is the Lagrange multiplier and $c$ is the constant in the constraint. 3. **Important rules:** - Take partial derivatives of $\mathcal{L}$ with respect to each variable and $\lambda$. - Set these derivatives equal to zero to form a system of equations. - Solve the system to find critical points. 4. **Intermediate work:** - Compute $$\frac{\partial \mathcal{L}}{\partial x} = 0, \quad \frac{\partial \mathcal{L}}{\partial y} = 0, \quad \frac{\partial \mathcal{L}}{\partial \lambda} = 0$$ - Simplify and solve these equations step-by-step. 5. **Explanation:** This method finds points where the gradient of $f$ is parallel to the gradient of $g$, which are candidates for maxima or minima under the constraint. Since the user did not provide specific functions $f$ and $g$, this is the general approach to solve such problems using Lagrangian functions.