Let f be a function defined by $f(x) = 2x^3 - 3x^2 - 36x + 2$, then which of the following are correct?

(A) The critical points of f(x) are -2 and 3.

(B) The function f(x) increases in the interval $(3, \infty)$

(C) The function f(x) decreases in the interval (-2,3)

(D) The function f(x) increases in the interval (-2,3)

Choose the correct answer from the options given below:

Given $f(x) = 2x^3 - 3x^2 - 36x + 2$

To find critical points, find where $f'(x) = 0$.

$f'(x) = 6x^2 - 6x - 36$

$6x^2 - 6x - 36 = 0$

$x^2 - x - 6 = 0$

$(x - 3)(x + 2) = 0$

$x = 3$ or $x = -2$

Statement (A) is correct: The critical points are -2 and 3.

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The critical points divide the number line into three intervals: $(-\infty, -2)$, $(-2, 3)$, and $(3, \infty)$.

For $f'(x) > 0$, the function increases. For $f'(x) < 0$, the function decreases.

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For interval $(-\infty, -2)$, test $x = -3$:

$f'(-3) = 6(-3)^2 - 6(-3) - 36$

$= 54 + 18 - 36$

$= 36 > 0$

The function increases on $(-\infty, -2)$.

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For interval $(-2, 3)$, test $x = 0$:

$f'(0) = 6(0)^2 - 6(0) - 36$

$= -36 < 0$

The function decreases on $(-2, 3)$.

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For interval $(3, \infty)$, test $x = 4$:

$f'(4) = 6(16) - 6(4) - 36$

$= 96 - 24 - 36$

$= 36 > 0$

The function increases on $(3, \infty)$.

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Statement (B) is correct: The function increases in $(3, \infty)$.

Statement (C) is correct: The function decreases in $(-2, 3)$.

Statement (D) is incorrect: The function decreases (not increases) in $(-2, 3)$.

Therefore, the correct statements are (A), (B), and (C).