If \( f(x)=(x+3)e^{-x} \) , then which of the following statements are True?
(A) Function \( f(x) \) is increasing on (-2, \( \infty \)).
(B) Function \( f(x) \) is decreasing on (-2, \( \infty \)).
(C) Function \( f(x) \) is increasing on (\( -\infty \), -2).
(D) Function \( f(x) \) is decreasing on (2, 3).
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B), (C) and (D) only
By the product rule:
\( \displaystyle f'(x)=e^{-x}-(x+3)e^{-x} \)
\( \displaystyle =-(x+2)e^{-x} \)
Since \( e^{-x}\gt 0 \) always, the sign of f'(x) is the sign of \( -(x+2) \):
\( \displaystyle x\lt -2:\ f'(x)\gt 0 \)
\( \displaystyle x\gt -2:\ f'(x)\lt 0 \)
So f is increasing on (−∞, −2) and decreasing on (−2, ∞).
Checking the statements
(A) Function \( f(x) \) is increasing on (-2, \( \infty \)). — Incorrect. f'(x) < 0 for x > −2, so f is decreasing there.
(B) Function \( f(x) \) is decreasing on (-2, \( \infty \)). — Correct. f'(x) < 0 for x > −2.
(C) Function \( f(x) \) is increasing on (\( -\infty \), -2). — Correct. f'(x) > 0 for x < −2.
(D) Function \( f(x) \) is decreasing on (2, 3). — Correct. (2, 3) lies inside (−2, ∞), where f is decreasing.