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Home » Thus, \( B = -1 \). So, - Feedz API

Thus, \( B = -1 \). So, - Feedz API

📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 Question:** A deep-sea microbial ecologist models the growth of bacteria near hydrothermal vents with the function \( f(x) = 5x^3 - 15x^2 + 10x \). Determine the critical points of \( f(x) \) to analyze potential growth spurts.

📌 Solution:** To find the critical points of \( f(x) = 5x^3 - 15x^2 + 10x \), we first compute its derivative:

📌 f'(x) = rac{d}{dx}(5x^3 - 15x^2 + 10x) = 15x^2 - 30x + 10

📌 Next, we set the derivative equal to zero to find the critical points:

📌 15x^2 - 30x + 10 = 0

📌 Divide the entire equation by 5:

📌 Using the quadratic formula \( x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 3 \), \( b = -6 \), and \( c = 2 \):

📌 x = rac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = rac{6 \pm \sqrt{36 - 24}}{6} = rac{6 \pm \sqrt{12}}{6}

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