A function is given by f(x) = x^3 - 6x^2 + 9x + 2. At which value of x does f(x) attain a local maximum?

Answer and explanation

Correct answer: D. x = 1

Compute the first derivative: f'(x) = 3x^2 - 12x + 9 = 3(x^2 - 4x + 3) = 3(x - 1)(x - 3). Critical points are at x = 1 and x = 3. Compute the second derivative: f''(x) = 6x - 12. At x = 1: f''(1) = 6(1) - 12 = -6 < 0 (concave down, local maximum). At x = 3: f''(3) = 6(3) - 12 = +6 > 0 (local minimum). Hence, local maximum is at x = 1.

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