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speed of sound in miles per hour - AssociationVoting

📅 May 18, 2026 👤 scraface

May 18, 2026

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📌 Thus, the value of \( k \) that ensures the minimum cost is zero is \(\boxed{12}\).

📌 An AI programmer is optimizing a model's performance using a quadratic cost function \( f(x) = ax^2 + bx + c \). If the vertex of this parabola corresponds to the minimum point at \( x = 3 \) and the cost at the vertex is \(-5\), find the relationship between \( a \) and \( c \) given \( b = -6 \).

📌 The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. Given \( h = 3 \) and \( k = -5 \), we have:

📌 f(x) = a(x - 3)^2 - 5

📌 We are also given \( b = -6 \). We rewrite the standard form \( f(x) = ax^2 + bx + c \) in vertex form by expanding:

📌 f(x) = a(x^2 - 6x + 9) - 5 = ax^2 - 6ax + 9a - 5

📌 Comparing coefficients with \( ax^2 + bx + c \), we identify:

📌 \( b = -6a \)

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