For a function \( f(x)=-x^2-2x+30 \) , which of the following statements are TRUE?
(A) \( f(x) \) is increasing on \( (-\infty,-1) \)
(B) \( f(x) \) is increasing on \( (-1,\infty) \)
(C) \( f(x) \) is decreasing on \( (-\infty,-1) \)
(D) \( f(x) \) is decreasing on \( (-1,\infty) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (D) only
\( \displaystyle f'(x)=-2x-2 \)
\( \displaystyle =-2(x+1) \)
So \( f'(x)\gt 0 \) on \( (-\infty,-1) \) and \( f'(x)\lt 0 \) on \( (-1,\infty) \): f is increasing on \( (-\infty,-1) \) and decreasing on \( (-1,\infty) \).
Checking the statements
(A) \( f(x) \) is increasing on \( (-\infty,-1) \) — Correct. For x < −1, \( x+1\lt 0 \), so \( f'(x)\gt 0 \).
(B) \( f(x) \) is increasing on \( (-1,\infty) \) — Incorrect. For x > −1, \( f'(x)\lt 0 \), so f is decreasing there.
(C) \( f(x) \) is decreasing on \( (-\infty,-1) \) — Incorrect. For x < −1, \( f'(x)\gt 0 \), so f is increasing there.
(D) \( f(x) \) is decreasing on \( (-1,\infty) \) — Correct. For x > −1, \( x+1\gt 0 \), so \( f'(x)\lt 0 \).