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Consider the function f: R → R defined as follows: f(x) = { x⁵ + 5x⁴ + 10x³ + 10x² + 3x + 1, when x < 0; x² - x + 1, for 0 ≤ x < 1; (2/3)x³ - 4x² + 7x - 8/3, for 1 ≤ x < 3; (x - 2)ln(x - 2) - x + 10/3, when x ≥ 3 }. Which of the following statements about f is/are true? (1) f is strictly increasing for x in (-∞, 0) (2) f is surjective (3) f' achieves a local maximum at x = 1 (4) f' is not continuous at x = 1.

1. f is strictly increasing for x in (-∞, 0)
2. f is surjective
3. f' achieves a local maximum at x = 1
4. f' is not continuous at x = 1

Correct answer: f' achieves a local maximum at x = 1

Solution

The derivative f'(x) achieves a local maximum at x = 1, as shown by analyzing the function's piecewise definition and checking the behavior of f'(x) around x = 1.

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