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Home » The Boys Next Season: Are They Starting Something BIG or Hiding Secrets? - Upplift

The Boys Next Season: Are They Starting Something BIG or Hiding Secrets? - Upplift

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 C''(2) = e^{-2}(5(2)^2 - 20(2) + 10) = e^{-2}(20 - 40 + 10) = e^{-2}(-10) < 0 \quad ( ext{local maximum}).

📌 Therefore, the concentration is maximized at \( x = 2 \).

📌 A historian of science is examining the impact of a scientific discovery over time, modeled by \( I(t) = rac{t^2}{1 + t^3} \), where \( t \) is time in centuries. Find the time \( t \) when the impact rate is increasing most rapidly.

📌 To find when the impact rate is increasing most rapidly, we need the maximum of the derivative of \( I(t) \):

📌 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:

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