Question

The function f defined by \(\mathrm{f}(x)=(x+2) \mathrm{e}^{-x}\) is

• A. decreasing for all \(x\in \mathbb{R}\)
• B. decreasing in \((-\infty ,-1)\) and increasing in ( \(-1,\infty\) )
• C. decreasing in \((-1,\infty )\) and increasing in ( \(-\infty ,-1\) )
• D. increasing for all \(x\in \mathbb{R}\)

Hint:

Differentiate and check the sign of the derivative on each side of the critical point.

Answer 1

The Correct Option is C

Step 1: Critical Point
$f'(x)=0$ gives $x=-1$.

Step 2: Test Values
At $x=-2$: $f'(-2)=e^{2} \gt 0$. At $x=0$: $f'(0)=-1 \lt 0$.

Step 3: Compare the options
The function rises before $-1$ and falls after it, which is option (C).

Final Answer:
The function increases up to $x=-1$ and decreases afterwards. \[ \boxed{\text{(C) }\text{decreasing in }(-1,\infty)\text{ and increasing in }(-\infty,-1)} \]