Function \(f(x) = x^3 - 3x + 3\) is
(A) Increasing in the interval \((-1, 1)\)
(B) Increasing in the interval \((1, \infty)\)
(C) Decreasing in the interval \((-1, 1)\)
(D) Increasing in the interval \((-\infty, -1) \cup (1, \infty)\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B), (C) and (D) only
The sign of the derivative decides where f increases or decreases:
\( \displaystyle f'(x) = 3x^2 - 3 = 3(x - 1)(x + 1) \)
\( \displaystyle f'(x) \gt 0 \text{ for } x \lt -1 \text{ and for } x \gt 1 \)
\( \displaystyle f'(x) \lt 0 \text{ for } -1 \lt x \lt 1 \)
So f is increasing in \((-\infty, -1)\) and in \((1, \infty)\), and decreasing in \((-1, 1)\).
Checking the statements
(A) Increasing in the interval \((-1, 1)\) — Incorrect. f is decreasing in \((-1, 1)\).
(B) Increasing in the interval \((1, \infty)\) — Correct. \(f'(x) \gt 0\) there.
(C) Decreasing in the interval \((-1, 1)\) — Correct. \(f'(x) \lt 0\) there.
(D) Increasing in the interval \((-\infty, -1) \cup (1, \infty)\) — Correct. f is increasing in each of the two intervals; the union is read interval by interval. (Taken as one set it would fail, since \(f(-1.5) = 4.125 \gt f(1.5) = 1.875\).)