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A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?
Interval
(a) (-∞, ∞) — x³ - 3x² + 3x + 3
(b) [2, ∞) — 2x³ - 3x² - 12x + 6
(c) (-∞, 1/3] — 3x² - 2x + 1
(d) (-∞, -4) — x³ + 6x² + 6
- (a) (-∞, ∞) — x³ - 3x² + 3x + 3
- (b) [2, ∞) — 2x³ - 3x² - 12x + 6
- (c) (-∞, 1/3] — 3x² - 2x + 1
- (d) (-∞, -4) — x³ + 6x² + 6
Correct answer: (c) (-∞, 1/3] — 3x² - 2x + 1
Solution
For 3x^2-2x+1, f'=6x-2 >= 0 only when x >= 1/3, so the function increases on [1/3, inf) and decreases on (-inf, 1/3]. Thus matching it to (-inf, 1/3] is incorrect; pairs (a), (b), (d) are all correctly matched.
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