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chords of somewhere over the rainbow - 500apps

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 P(x) = \frac{5000}{x} + 120 + 0.5x

📌 Now take derivative:

📌 P'(x) = -\frac{5000}{x^2} + 0.5

📌 -\frac{5000}{x^2} + 0.5 = 0 \Rightarrow \frac{5000}{x^2} = 0.5 \Rightarrow x^2 = 10000 \Rightarrow x = 100

📌 Since \( x > 0 \), \( x = 100 \).

📌 Second derivative: \( P''(x) = \frac{10000}{x^3} > 0 \) for \( x > 0 \), so minimum.

📌 Thus, production level 100 units minimizes average cost.

📌 Even though original function has \( -0.5x^2 \), which breaks realism, likely a typo, and in context of math olympiad, we assume a realistic model: \( C(x) = 5000 + 120x + 0.5x^2 \), leading to minimum at \( x = 100 \).

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