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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Thus, the value of \(g(f(4))\) is \(\boxed{101}\).

📌 Question:** A chemical engineer is modeling biofuel yield based on reactant concentrations. If the yield \(Y\) is given by \(Y = 4x^2 - 12x + 9\), compute the minimum possible yield and the value of \(x\) at which it occurs.

📌 The expression \(Y = 4x^2 - 12x + 9\) is a quadratic in standard form. Since the coefficient of \(x^2\) is positive, the parabola opens upwards, and the minimum occurs at the vertex.

📌 The vertex of a parabola \(ax^2 + bx + c\) occurs at:

📌 x = -\frac{b}{2a} = -\frac{-12}{2 \cdot 4} = \frac{12}{8} = \frac{3}{2}.

📌 Substitute \(x = \frac{3}{2}\) into the equation to find the minimum yield:

📌 Y = 4\left(\frac{3}{2}\right)^2 - 12\left(\frac{3}{2}\right) + 9 = 4\left(\frac{9}{4}\right) - 18 + 9 = 9 - 18 + 9 = 0.

📌 Thus, the minimum yield is \(0\) at \(x = \frac{3}{2}\), and the minimum value is \(\boxed{0}\).

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