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📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 To find the critical points, we compute the derivative \( g'(t) \) and set it to zero.

📌 g(t) = t^3 - 4t^2 + 4t

📌 g'(t) = 3t^2 - 8t + 4

📌 Setting \( g'(t) = 0 \) gives the equation \( 3t^2 - 8t + 4 = 0 \). Solving this quadratic equation using the quadratic formula \( t = rac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 3 \), \( b = -8 \), and \( c = 4 \), we find:

📌 t = rac{8 \pm \sqrt{(-8)^2 - 4 \cdot 3 \cdot 4}}{2 \cdot 3} = rac{8 \pm \sqrt{64 - 48}}{6} = rac{8 \pm \sqrt{16}}{6} = rac{8 \pm 4}{6}

📌 The solutions are \( t = 2 \) and \( t = rac{2}{3} \).

📌 oxed{2, rac{2}{3}}

📌 An industrial hygienist is evaluating exposure levels to a new chemical in a factory, modeled by a polynomial \( h(x) = 2x^3 - 9x^2 + 12x - 4 \). Calculate the product of the non-zero roots of this polynomial.

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