For the function \( f(x)=x^2e^{-x} \), which of the following statement are TRUE?
(A) \( f(x) \) is decreasing on \( (-\infty,0) \)
(B) \( f(x) \) is increasing on \( \mathrm{R} \)
(C) \( f(x) \) is increasing on (0, 2)
(D) \( f(x) \) is decreasing on \( (2,\infty) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
\( \displaystyle f'(x)=2xe^{-x}-x^2e^{-x} \)
\( \displaystyle =x(2-x)e^{-x} \)
Since \( e^{-x}\gt 0 \), the sign of f′(x) is the sign of x(2 − x):
\( \displaystyle x\lt 0:\ (-)(+)\lt 0 \)
\( \displaystyle 0\lt x\lt 2:\ (+)(+)\gt 0 \)
\( \displaystyle x\gt 2:\ (+)(-)\lt 0 \)
So f′(x) < 0 on \( (-\infty,0) \) and on \( (2,\infty) \), and f′(x) > 0 on (0, 2).
Checking the statements
(A) \( f(x) \) is decreasing on \( (-\infty,0) \) — Correct. f′(x) < 0 there.
(B) \( f(x) \) is increasing on \( \mathrm{R} \) — Incorrect. f decreases on \( (-\infty,0) \) and on \( (2,\infty) \).
(C) \( f(x) \) is increasing on (0, 2) — Correct. f′(x) > 0 there.
(D) \( f(x) \) is decreasing on \( (2,\infty) \) — Correct. f′(x) < 0 there.