The function \( f(x)=-2x^{3}-9x^{2}-12x+5 \) is decreasing in the interval/(s) :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( (-\infty,-2)\cup(-1,\infty) \)
A function is decreasing where \( f'(x)\lt 0 \). Here
\( f'(x)=-6x^{2}-18x-12 \)
\( =-6(x+1)(x+2) \)
So \( f'(x)\lt 0 \) when \( (x+1)(x+2)\gt 0 \).
This holds when both factors are positive (\( x\gt -1 \)) or both are negative (\( x\lt -2 \)).
Hence \( f \) is decreasing on \( (-\infty,-2)\cup(-1,\infty) \).
Why the other options are incorrect
Option (1) \( (-\infty,-2)\cup(-2,-1) \) — Incorrect. It includes \( (-2,-1) \), where \( f'(x)\gt 0 \).
Option (2) \( (-2,-1) \) — Incorrect. This is the interval where \( f \) is increasing.
Option (4) \( (-\infty,\infty) \) — Incorrect. \( f'(x) \) is positive between \( -2 \) and \( -1 \), so \( f \) is not decreasing on the whole line.