Question

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 \( (-1, \infty) \)

• (B) and (D) only
• (B) and (C) only
• (A) and (C) only
• (A) and (D) only

Hint:

For a downward-opening parabola \( ax^2 + bx + c \) (\( a < 0 \)), the function increases until the vertex \( x = -b/2a \) and decreases thereafter.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To determine where a function is increasing or decreasing, we analyze the sign of its first derivative, \( f'(x) \).

Step 2: Detailed Explanation:
Calculate the derivative: \( f'(x) = \frac{d}{dx}(-x^2 - 2x + 30) = -2x - 2 \).
Set \( f'(x) = 0 \) to find the critical point: \( -2x - 2 = 0 \implies x = -1 \).
For \( x < -1 \), choose \( x = -2 \): \( f'(-2) = -2(-2) - 2 = 4 - 2 = 2 > 0 \). Thus, \( f(x) \) is increasing on \( (-\infty, -1) \).
For \( x > -1 \), choose \( x = 0 \): \( f'(0) = -2(0) - 2 = -2 < 0 \). Thus, \( f(x) \) is decreasing on \( (-1, \infty) \).
Comparing with options, (A) and (D) are true.

Step 3: Final Answer:
The correct statements are (A) and (D).