Set:

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• Given the confusion, let's assume the function is meant to be minimized as total cost, but the problem says "cost per unit".
• Alternatively, perhaps "cost per unit" is misphrased, and we minimize average cost, but as shown, it has no minimum.
• But in business models, average cost \( C(x)/x = 5000/x + 120 - 0.5x \) has a minimum when derivative is zero.
• Set \( P'(x) = 0 \): no solution. So minimum does not exist — function decreases for large \( x \)? No, \( -0.5x \) dominates, so \( P(x) \to -\infty \), impossible.
• This suggests the model is invalid, but for math olympiad, likely the function is \( C(x) = 5000 + 120x + 0.5x^2 \). But it says \( -0.5x^2 \).
• Wait — perhaps it's \( C(x) = 5000 + 120x - 0.5x^2 \), but \( x \) has an upper limit. But no.

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