A company’s production cost, in dollars, is modeled by the function \(C(x) = 2x^3 – 15x^2 + 36x + 40\), where \(x\) represents the number of units produced (in hundreds). The company wants to determine the production level that minimizes cost.
Answer the following:
(a) Find the value(s) of \(x\) for which the first derivative equals zero.
(b) Evaluate the second derivative at the critical points to determine which corresponds to a minimum.
(c) Interpret the result in the context of production optimization.
Question
Differentiate \(C(x) = 2x^3 – 15x^2 + 36x + 40\) with respect to \(x\) and solve \(C'(x)=0\).
Hint
Hint
Hint
Differentiate term-by-term and factor the resulting quadratic.
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- Formula Booklet
Complete Answer
The derivative is \(C'(x)=6x^2 – 30x + 36\). Setting \(C'(x)=0\) gives \(x^2 – 5x + 6=0\), so \(x=2\) and \(x=3\).
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Question
Compute the second derivative \(C”(x)\) and evaluate it at \(x=2\) and \(x=3\). Indicate which critical point gives a minimum cost.
Hint
Hint
Hint
Differentiate \(C'(x)\) to find \(C”(x)\) and use the sign of \(C”(x)\) to determine the nature of the extrema.
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Complete Answer
The second derivative is \(C”(x)=12x-30\). Evaluating gives \(C”(2)=-6\) (indicating a maximum) and \(C”(3)=6\) (indicating a minimum). Thus, \(x=3\) minimizes cost.
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Question
Explain the significance of the result \(x=3\) in the context of the production cost model, given that \(x\) is measured in hundreds of units.
Hint
Hint
Hint
Convert the value of \(x\) into actual production units and relate it to minimizing cost.
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Explanation
- Formula Booklet
Complete Answer
The result \(x=3\) means that the minimum cost is achieved when 300 units are produced, making 300 the most cost-efficient production level.
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