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Home » \sum_{\substacka=1\\b=0\\c=0}^2,1,1 2^a \cdot 3^b \cdot 5^c - DNSFLEX

\sum_{\substacka=1\\b=0\\c=0}^2,1,1 2^a \cdot 3^b \cdot 5^c - DNSFLEX

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 t = rac{-4 \pm \sqrt{16 + 144}}{6} = rac{-4 \pm \sqrt{160}}{6} = rac{-4 \pm 4\sqrt{10}}{6} = rac{-2 \pm 2\sqrt{10}}{3}

📌 So the critical points are:

📌 t_1 = rac{-2 - 2\sqrt{10}}{3}, \quad t_2 = rac{-2 + 2\sqrt{10}}{3}

📌 Since the denominator of $ C'(t) $ is always positive, the sign of $ C'(t) $ depends on the numerator. The expression $ -3t^2 - 4t + 12 $ is positive between the roots. Therefore, $ C(t) $ is increasing for:

📌 oxed{ rac{-2 - 2\sqrt{10}}{3} < t < rac{-2 + 2\sqrt{10}}{3} }

📌 Question: A sustainable materials researcher models the degradation rate of a biodegradable polymer with the function $ D(t) = rac{t^2 + 1}{t - 2} $. Find all vertical asymptotes and holes in the graph of $ D(t) $.

📌 Solution: To find vertical asymptotes and holes, we examine the denominator of $ D(t) = rac{t^2 + 1}{t - 2} $.

📌 t - 2 = 0 \Rightarrow t = 2

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