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:

To determine where $f(x) = x^3 - 3x + 3$ is increasing or decreasing, the derivative indicates the slope at any point. Positive slope means increasing, negative slope means decreasing.

$f'(x) = 3x^2 - 3$

$f'(x) = 3(x^2 - 1)$

$f'(x) = 3(x - 1)(x + 1)$

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Setting $f'(x) = 0$:

$3(x - 1)(x + 1) = 0$

Critical points: $x = -1$ and $x = 1$

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Testing the sign of $f'(x)$ in each interval:

For $(-\infty, -1)$, test $x = -2$:

$f'(-2) = 3(-2 - 1)(-2 + 1)$

$= 3(-3)(-1)$

$= 9 > 0$

Function is increasing.

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For $(-1, 1)$, test $x = 0$:

$f'(0) = 3(0 - 1)(0 + 1)$

$= 3(-1)(1)$

$= -3 < 0$

Function is decreasing.

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For $(1, \infty)$, test $x = 2$:

$f'(2) = 3(2 - 1)(2 + 1)$

$= 3(1)(3)$

$= 9 > 0$

Function is increasing.

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Evaluating each option:

(A) Increasing in $(-1, 1)$: False (the function is decreasing)

(B) Increasing in $(1, \infty)$: True

(C) Decreasing in $(-1, 1)$: True

(D) Increasing in $(-\infty, -1) \cup (1, \infty)$: True

The correct statements are (B), (C), and (D).