For the function $f(x) = -2x^3+3x^2+36x-10$, which of the following is/are true?

(A) f is increasing in (-∞, -2)
(B) f is increasing in (-2, 3)
(C) f is decreasing in (-∞, -2)
(D) f is decreasing in (3, ∞)

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:

$f(x) = -2x^{3} + 3x^{2} + 36x - 10$

Compute derivative:

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

Factor:

$f'(x) = -6(x^{2} - x - 6)$

$= -6(x - 3)(x + 2)$

Now check the sign of $f'(x)$ in intervals:

Critical points: $x = -2,\; x = 3$

Interval $(-\infty, -2)$:

Choose $x = -3$ → $(x - 3)(x + 2) = (-6)(-1) = 6$ → $f'(x) = -6(6) < 0$

→ decreasing

Interval $(-2, 3)$:

Choose $x = 0$ → $(x - 3)(x + 2) = (-3)(2) = -6$ → $f'(x) = -6(-6) > 0$

→ increasing

Interval $(3, \infty)$:

Choose $x = 4$ → $(x - 3)(x + 2) = (1)(6) = 6$ → $f'(x) = -6(6) < 0$

→ decreasing

Correct statements:

(C) $f$ is decreasing in $(-\infty,\,-2)$ ✔

(B) $f$ is increasing in $(-2,\,3)$ ✔

(D) $f$ is decreasing in $(3,\,\infty)$ ✔

Correct options: (B), (C), (D).