Question

The function $f(x) = (x + 2)e^{-x}$ is

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

Hint:

Sign analysis of the first derivative helps determine intervals of increase and decrease.

Answer 1

The Correct Option is C

Step 1: Differentiating the function.
\[ f(x) = (x + 2)e^{-x} \] \[ f'(x) = (x + 2)(-e^{-x}) + e^{-x} = -e^{-x}(x + 1) \]
Step 2: Finding critical points.
\[ f'(x) = 0 \Rightarrow x = -1 \]
Step 3: Sign of the derivative.
For $x<-1$, $f'(x)>0$ so the function is increasing.
For $x>-1$, $f'(x)<0$ so the function is decreasing.
Step 4: Conclusion.
The function increases in $(-\infty,-1)$ and decreases in $(-1,\infty)$.