Cost Optimization Using Derivatives

00:00 6 marks

A manufacturing company models the cost of producing \(x\) units of a product using the function \(C(x) = ax^3 – bx^2 + cx + d\), where \(a\), \(b\), \(c\), and \(d\) are positive constants and \(x \ge 0\) represents the number of units produced.

(a) Find the derivative of the cost function, \(C'(x)\).

(b) Explain how to find the production level \(x\) that minimizes the cost using calculus. Include the steps to verify that this production level corresponds to a minimum using the second derivative test.

Hint Hint

Hint

For part (b), set the first derivative equal to zero to find critical points, and then use the second derivative to determine if it’s a minimum.

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Complete Answer

(a) \(C'(x) = 3ax^2 – 2bx + c\)
(b) Set \(C'(x) = 0\) and solve for \(x\). Calculate \(C”(x) = 6ax – 2b\). If for a critical point \(x_0\), \(C”(x_0) > 0\), then \(x_0\) is a local minimum.

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Answer Explanation Answer Explanation

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Common Errors Common Errors

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