The interval in which the function \( f(x)=-x^2-2x+18 \) is decreasing is :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( (-1,\infty) \)
A function is decreasing where its derivative is negative.
\( f'(x)=-2x-2 \)
\( =-2(x+1) \)
\( f'(x)\lt 0 \) when x + 1 > 0, i.e. x > −1. So f is decreasing on \( (-1,\infty) \).
Why the other options are incorrect
Option (1) \( (-\infty,-1) \) — Incorrect. Here \( f'(x)\gt 0 \), so f is increasing on this interval.
Option (2) \( (-3,\infty) \) — Incorrect. On the part from −3 to −1, \( f'(x)\gt 0 \), so f is not decreasing on the whole interval.
Option (4) \( (-\infty,-2] \) — Incorrect. Here x < −1, so \( f'(x)\gt 0 \) and f is increasing.