Function $f(x) = x^3 − 3x + 3$ is
(A) Increasing in the interval (-1, 1)
(B) Increasing in the interval (1, ∞)
(C) Decreasing in the interval (-1, 1)
(D) Increasing in the interval (-∞, -1)∪(1,∞)
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
Given function:
\(f(x) = x^3 - 3x + 3\)
Find the derivative:
\[ f'(x) = 3x^2 - 3 = 3(x^2 - 1) \]
Factor derivative:
\[ f'(x) = 3(x - 1)(x + 1) \]
Set \(f'(x) = 0\) to find critical points:
\[ 3(x - 1)(x + 1) = 0 => x = -1, 1 \]
Check the behavior of \(f'(x)\) around critical points by evaluating the sign:
| Interval | Test point | \(f'(x)\) sign | Behavior of \(f(x)\) |
|---|---|---|---|
| \((-\infty, -1)\) | \(-2\) | \(f'(-2) = 3(4 - 1) = 9 > 0\) | Increasing |
| \((-1, 1)\) | \(0\) | \(f'(0) = 3(0 - 1) = -3 < 0\) | Decreasing |
| \((1, \infty)\) | \(2\) | \(f'(2) = 3(4 - 1) = 9 > 0\) | Increasing |
Conclusion:
- Function is increasing on \((-\infty, -1)\) and \((1, \infty)\)
- Function is decreasing on \((-1, 1)\)
Thus, correct options are:
(B), (C), and (D)