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: Understanding the Concept
A function increases where \(f'(x) \gt 0\) and decreases where \(f'(x) \lt 0\).

Step 2: Differentiate
\(f'(x)=e^{-x}-(x+2)e^{-x}=-(x+1)e^{-x}\).

Step 3: Sign of f'
Since \(e^{-x} \gt 0\), the sign of \(f'\) is the sign of \(-(x+1)\).
For \(x \lt -1\), \(f' \gt 0\): increasing. For \(x \gt -1\), \(f' \lt 0\): decreasing.

Step 4: Result
The function increases in \((-\infty,-1)\) and decreases in \((-1,\infty)\), option (C).

Final Answer:
It is increasing for \(x \lt -1\) and decreasing for \(x \gt -1\). \[ \boxed{\text{(C) }\text{decreasing in }(-1,\infty)\text{ and increasing in }(-\infty,-1)} \]