1. The problem is to find the marginal products $C^*$ and $L^#$ for the production function $$Q(C, L) = C^{\frac{1}{2}} \cdot L^{\frac{1}{2}}.$$ 2. The marginal product of input $C$ (denoted $C^*$) is the partial derivative of $Q$ with respect to $C$: $$C^* = \frac{\partial Q}{\partial C} = \frac{\partial}{\partial C} \left(C^{\frac{1}{2}} L^{\frac{1}{2}}\right).$$ Using the power rule and treating $L$ as a constant: $$C^* = \frac{1}{2} C^{-\frac{1}{2}} L^{\frac{1}{2}} = \frac{1}{2} \frac{L^{\frac{1}{2}}}{C^{\frac{1}{2}}}.$$ 3. The marginal product of input $L$ (denoted $L^#$) is the partial derivative of $Q$ with respect to $L$: $$L^# = \frac{\partial Q}{\partial L} = \frac{\partial}{\partial L} \left(C^{\frac{1}{2}} L^{\frac{1}{2}}\right).$$ Using the power rule and treating $C$ as a constant: $$L^# = \frac{1}{2} C^{\frac{1}{2}} L^{-\frac{1}{2}} = \frac{1}{2} \frac{C^{\frac{1}{2}}}{L^{\frac{1}{2}}}.$$ 4. Summary: - Marginal product of $C$: $$C^* = \frac{1}{2} \frac{L^{\frac{1}{2}}}{C^{\frac{1}{2}}}.$$ - Marginal product of $L$: $$L^# = \frac{1}{2} \frac{C^{\frac{1}{2}}}{L^{\frac{1}{2}}}.$$ These represent how output $Q$ changes with a small increase in $C$ or $L$ respectively, holding the other input constant.