Optimizing Cost Function for Manufacturing

00:00 6 marks

A manufacturing company has determined that its production cost (in thousand dollars) for producing x hundred units is given by the function \(C(x) = -0.02x^3 + 1.2x^2 – 20x + 500\).

(a) Find the stationary points of \(C(x)\) by differentiating and setting \(C'(x)=0\).

(b) Apply the second derivative test to classify each stationary point as a local maximum or minimum, and determine the production level that minimizes the cost.

Assume that \(x\) represents the number of hundred units produced and is within the feasible production range.

Hint Hint

Hint

Differentiate \(C(x)\) to obtain \(C'(x)\), solve \(C'(x)=0\), then use \(C”(x)\) to test the nature of each stationary point.

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation
Examiner Tips Examiner Tips
Common Errors Common Errors
Real World Context Real World Context
Focus Focus
Formula Booklet

Answer Answer

Complete Answer

Stationary points occur at approximately \(x \approx 11.8\) (local minimum) and \(x \approx 28.2\) (local maximum); the cost is minimized at \(x \approx 11.8\) hundred units.

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Examiner Tips Examiner Tips

Common Errors Common Errors

Focus Focus

Formula Booklet