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ight) = \sin\left(rac\pi2 - wispro

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 f(t) = \frac{e^{-0.5t}}{(1 + 9e^{-0.5t})^2}.

📌 Let $ u = e^{-0.5t} $, so $ f(t) = \frac{u}{(1 + 9u)^2} $. Now find the maximum of this function with respect to $ u > 0 $.

📌 f(u) = \frac{u}{(1 + 9u)^2}.

📌 Differentiate using the quotient rule:

📌 f'(u) = \frac{(1 + 9u)^2 \cdot 1 - u \cdot 2(1 + 9u)(9)}{(1 + 9u)^4}.

📌 Simplify numerator:

📌 f'(u) = \frac{(1 + 9u)(1 - 9u)}{(1 + 9u)^4} = \frac{1 - 81u^2}{(1 + 9u)^3}.

📌 Set $ f'(u) = 0 \Rightarrow 1 - 81u^2 = 0 \Rightarrow u^2 = \frac{1}{81} \Rightarrow u = \frac{1}{9} $ (since $ u > 0 $).

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