1. **Problem:** Given the total-cost function $$C = Q^3 - 5Q^2 + 14Q + 75$$, write out the variable-cost (VC) function, find its derivative, and interpret the economic meaning of that derivative. 2. **Step 1: Identify the variable-cost function.** The total cost $$C$$ consists of fixed costs and variable costs. Fixed costs do not depend on $$Q$$, while variable costs do. 3. **Step 2: Extract fixed cost and variable cost.** The fixed cost is the constant term: $$75$$. Variable cost function $$VC(Q)$$ is total cost minus fixed cost: $$VC = Q^3 - 5Q^2 + 14Q$$ 4. **Step 3: Find the derivative of the variable-cost function.** The derivative $$\frac{dVC}{dQ}$$ is: $$\frac{dVC}{dQ} = 3Q^2 - 10Q + 14$$ 5. **Step 4: Economic interpretation.** The derivative of the variable cost with respect to quantity $$Q$$ is the marginal cost (MC), which represents the additional cost of producing one more unit of output. **Final answers:** - Variable-cost function: $$VC = Q^3 - 5Q^2 + 14Q$$ - Marginal cost (derivative of VC): $$MC = 3Q^2 - 10Q + 14$$ This means the cost to produce one additional unit changes with $$Q$$ according to the quadratic function above.