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now consider this the student brings - Coaching Toolbox

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 The function \( f(x) = x^3 - 6x^2 + 9x + c \) has a local maximum at \( x = 1 \). Find \( c \) if \( f(1) = 2 \).

📌 Solution 8:

📌 \( f'(x) = 3x^2 - 12x + 9 \).

📌 Setting \( f'(x) = 0 \) for critical points:

📌 3x^2 - 12x + 9 = 0 \Rightarrow x^2 - 4x + 3 = 0 \Rightarrow (x-3)(x-1) = 0

📌 So \( x = 1 \) and \( x = 3 \) are critical points.

📌 Check \( f(1) = 2 \):

📌 f(1) = 1^3 - 6(1)^2 + 9(1) + c = 1 - 6 + 9 + c = 4 + c

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