Alternatively, perhaps the function is \( C(x) = 5000 + 120x - 0.5x^2 \), and we want to minimize **total cost**, not per unit. But the problem says "cost per unit."
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- But assuming the function is correct as stated, and the goal is to minimize \( P(x) = \frac{5000}{x} + 120 - 0.5x \), set:
- P'(x) = -\frac{5000}{x^2} - 0.5 = 0 \quad \text{has no solution}
- So no minimum — but this contradicts realistic models.
- Reinterpreting: "cost per unit" is standard, and we must minimize \( P(x) = C(x)/x \). But if \( C(x) = 5000 + 120x - 0.5x^2 \), then for large \( x \), the \( -0.5x^2 \) dominates and cost per unit becomes negative — impossible.
- Therefore, likely the quadratic coefficient is positive. Assume a typo, and the intended model is \( C(x) = 5000 + 120x + 0.5x^2 \). But the problem says \( -0.5x^2 \).
- Alternatively, perhaps it's \( C(x) = 5000 + 120x + 0.5x^2 \)? But it says \( -0.5x^2 \).
