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📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 First, compute \( I'(t) \) using the quotient rule, where \( u = t^2 \) and \( v = 1 + t^3 \):

📌 I'(t) = rac{u'v - uv'}{v^2} = rac{(2t)(1 + t^3) - t^2(3t^2)}{(1 + t^3)^2}.

📌 I'(t) = rac{2t + 2t^4 - 3t^4}{(1 + t^3)^2} = rac{2t - t^4}{(1 + t^3)^2}.

📌 Now, find \( I''(t) \) to locate where \( I'(t) \) has a maximum:

📌 Let \( N(t) = 2t - t^4 \) and \( D(t) = (1 + t^3)^2 \), then:

📌 I''(t) = rac{N'(t)D(t) - N(t)D'(t)}{[D(t)]^2}.

📌 Compute derivatives:

📌 N'(t) = 2 - 4t^3, \quad D'(t) = 2(1 + t^3)(3t^2) = 6t^2(1 + t^3).

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