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Home » Compute the third derivative \( p'''(x) \): - Treasure Valley Movers

Compute the third derivative \( p'''(x) \): - Treasure Valley Movers

📅 April 16, 2026 👤 scraface

Apr 16, 2026

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📌 v\sqrt{v} - 5v + 6\sqrt{v} = 0, \quad v > 0

📌 Note that \( v\sqrt{v} = v^{3/2} \) and \( \sqrt{v} = v^{1/2} \). Let \( u = \sqrt{v} \), so \( v = u^2 \), and \( v^{3/2} = u^3 \). Substitute:

📌 u^3 - 5u^2 + 6u = 0

📌 u(u^2 - 5u + 6) = 0

📌 \Rightarrow u(u - 2)(u - 3) = 0

📌 Solutions: \( u = 0 \), \( u = 2 \), \( u = 3 \)

📌 But \( v > 0 \), so \( u = \sqrt{v} > 0 \), thus \( u = 0 \) is invalid. Valid solutions: \( u = 2 \Rightarrow v = 4 \), \( u = 3 \Rightarrow v = 9 \)

📌 Sum of real roots: \( 4 + 9 = 13 \)

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