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Let f(x) = x^3 − 3x^2 + 2x. For which x does f have criti... | QuestionStock

Representative image for question: Let f(x) = x^3 − 3x^2 + 2x. For which x does f have critical points (where f'(x) = 0)?

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Mathematics

Let f(x) = x^3 − 3x^2 + 2x. For which x does f have critical points (where f'(x) = 0)?

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Let f(x) = x^3 − 3x^2 + 2x. For which x does f have critical points (where f'(x) = 0)?

Options:

x = (3 ± √3)/3 (approximately 0.42265 and 1.57735)
x = 0 and x = 2
x = −1 and x = 3
No real critical points

Correct answer: x = (3 ± √3)/3 (approximately 0.42265 and 1.57735)

Explanation: f'(x)=3x^2−6x+2, whose discriminant is 12, giving roots x=(3±√3)/3 ≈ 0.42265 and 1.57735; the discriminant 12 yields 2√3 as the radical.

Created 2026-03-10T17:46:48.103Z. Updated 2026-03-10T17:46:48.103Z.