1. **Problem Statement:** We need to use the Lagrange multiplier method to find the quantities of three inputs that minimize the cost function subject to a given constraint. 2. **General Approach:** Suppose the cost function is $C(x,y,z)$ and the constraint is $g(x,y,z) = 0$. The Lagrange function is defined as: $$\mathcal{L}(x,y,z,\lambda) = C(x,y,z) + \lambda g(x,y,z)$$ where $\lambda$ is the Lagrange multiplier. 3. **Step-by-step:** - Take partial derivatives of $\mathcal{L}$ with respect to $x$, $y$, $z$, and $\lambda$: $$\frac{\partial \mathcal{L}}{\partial x} = 0, \quad \frac{\partial \mathcal{L}}{\partial y} = 0, \quad \frac{\partial \mathcal{L}}{\partial z} = 0, \quad \frac{\partial \mathcal{L}}{\partial \lambda} = 0$$ - Solve this system of equations simultaneously to find $x$, $y$, $z$, and $\lambda$. 4. **Important Rules:** - The constraint must be explicitly stated. - The cost function should be differentiable. - The solution must satisfy both the cost minimization and the constraint. 5. **Example (if given):** If the cost function is $C = w_1 x + w_2 y + w_3 z$ and the constraint is $f(x,y,z) = k$, then: $$\mathcal{L} = w_1 x + w_2 y + w_3 z + \lambda (f(x,y,z) - k)$$ 6. **Solve the system:** - Compute $\frac{\partial \mathcal{L}}{\partial x} = w_1 + \lambda \frac{\partial f}{\partial x} = 0$ - Compute $\frac{\partial \mathcal{L}}{\partial y} = w_2 + \lambda \frac{\partial f}{\partial y} = 0$ - Compute $\frac{\partial \mathcal{L}}{\partial z} = w_3 + \lambda \frac{\partial f}{\partial z} = 0$ - Use $f(x,y,z) = k$ to solve for $x,y,z$. 7. **Interpretation:** The values of $x,y,z$ found minimize the cost while satisfying the constraint. Since the exact cost function and constraint are not provided, this is the general method to solve such problems using Lagrange multipliers.