Profit Maximization using First Principles

00:00 7 marks

A firm’s profit function is given by \(P(x) = -x^3 + 6x^2 – 9x + 20\), where \(x\) is the production level in hundreds of units. Using the first principles definition of the derivative, find \(P'(x)\) and determine the production level that maximizes profit. Verify that this production level indeed yields a maximum profit using the second derivative test.

Hint Hint

Hint

Recall the definition of the derivative from first principles: \(P'(x) = \lim_{h \to 0}\frac{P(x+h)-P(x)}{h}\). Carefully expand \(P(x+h)\) and simplify the expression before taking the limit. Remember to use the second derivative test to confirm if the critical point is a maximum.

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Answer Answer

Complete Answer

\(P'(x) = -3x^2 + 12x – 9\) and maximum profit occurs at a production level of \(x = 3\) hundred units.

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