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:

(B), (C) and (D) only

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:

Conclusion:

• Function is increasing on \((-\infty, -1)\) and \((1, \infty)\)
• Function is decreasing on \((-1, 1)\)

Thus, correct options are:

(B), (C), and (D)