The interval in which \( f(x)=-2x^3-9x^2-12x+1 \) is increasing, is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( (-2,-1) \)
Since \( f(x)=-2x^3-9x^2-12x+1 \),
\( \displaystyle f'(x)=-6x^2-18x-12 \)
\( \displaystyle =-6(x^2+3x+2) \)
\( \displaystyle =-6(x+1)(x+2) \)
f is increasing where f'(x) > 0, i.e. where \( (x+1)(x+2)\lt 0 \) (the factor −6 reverses the sign).
The product of (x + 1) and (x + 2) is negative between the roots:
\( \displaystyle -2\lt x\lt -1 \)
So f is increasing on (−2, −1).
Why the other options are incorrect
Option (1) \( (-\infty,-2) \) — Incorrect. For x < −2, (x + 1)(x + 2) > 0, so f'(x) < 0 and f is decreasing there.
Option (2) \( (-1,\infty) \) — Incorrect. For x > −1, (x + 1)(x + 2) > 0, so f is decreasing there too.
Option (4) \( (-\infty,-2)\cup(-1,\infty) \) — Incorrect. This is where f is decreasing; the negative factor −6 has been ignored.
NCERT (2026-27 edition):
Increasing and Decreasing Functions
