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Home » Hair Bun Secrets That’ll Make Your Bun Look Like Hair Magic! - Upplift

Hair Bun Secrets That’ll Make Your Bun Look Like Hair Magic! - Upplift

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 To confirm it’s a maximum, check the sign of \( I'(t) \): decreasing before \( t = \sqrt{10} \), increasing after? Actually, the numerator \( -1000t^2 + 10000 \) is positive for \( t < \sqrt{10} \) and negative for \( t > \sqrt{10} \), so \( I'(t) \) changes from positive to negative — maximum at \( t = \sqrt{10} \).

📌 Thus, the infection rate peaks at \( t = \sqrt{10} \) days.

📌 \(\boxed{\sqrt{10}}\)

📌 Question:** A venture capitalist is evaluating a cleantech startup’s cost model given by \( C(x) = 5000 + 120x - 0.5x^2 \), where \( C \) is the cost in dollars for producing \( x \) units. Find the production level \( x \) that minimizes cost per unit.

📌 The cost per unit is \( P(x) = \frac{C(x)}{x} = \frac{5000 + 120x - 0.5x^2}{x} = \frac{5000}{x} + 120 - 0.5x \), for \( x > 0 \).

📌 To minimize \( P(x) \), take the derivative:

📌 P'(x) = -\frac{5000}{x^2} + 0 - 0.5 = -\frac{5000}{x^2} - 0.5

📌 Set \( P'(x) = 0 \):

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