To determine where the function $f(x)$ is increasing and decreasing, we need to analyze the sign of its derivative $f'(x)$.

Given $f'(x) = (x+2)e^{-x}$, we will:

1. Find the critical points where $f'(x) = 0$ or $f'(x)$ is undefined.
2. Determine the intervals where $f'(x)$ is positive or negative.

From the previous steps, we found that the critical point is $x = -2$.

Step 1: Critical Points

The critical point is $x = -2$.

Step 2: Sign Analysis of $f'(x)$

To determine where $f(x)$ is increasing or decreasing, we need to check the sign of $f'(x)$ on intervals determined by the critical point $x = -2$.

The intervals to consider are:

1. $(-\infty, -2)$
2. $(-2, \infty)$

Checking the Sign of $f'(x)$

Interval $(-\infty, -2)$:

Pick a test point, say $x = -3$: $f'(-3) = (-3+2)e^{-(-3)} = -1e^{3} = -e^{3}$ Since $-e^{3} < 0$, $f'(x) < 0$ on $(-\infty, -2)$. Thus, $f(x)$ is decreasing on this interval.

Interval $(-2, \infty)$:

Pick a test point, say $x = 0$: $f'(0) = (0+2)e^{-0} = 2e^{0} = 2$ Since $2 > 0$, $f'(x) > 0$ on $(-2, \infty)$. Thus, $f(x)$ is increasing on this interval.

Conclusion:

• $f(x)$ is decreasing on the interval $(-\infty, -2)$.
• $f(x)$ is increasing on the interval $(-2, \infty)$.

Would you like further details or have any questions?

Here are 5 related questions you might ask:

1. How do you find the intervals where a function is increasing or decreasing?
2. What are the critical points, and how do they relate to the function's behavior?
3. Can you explain the first derivative test in detail?
4. How can we use sign charts to determine where a function increases or decreases?
5. Can you provide an example of a function with multiple critical points and analyze its intervals?

Tip: When analyzing the sign of the derivative to determine where a function is increasing or decreasing, it is useful to choose test points within each interval separated by the critical points.