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Let's go through the given problem step by step.

### Given:
We are asked to analyze the function whose derivative is:

\[
f'(x) = (x + 4) e^{-x}
\]

### Part (a): **Critical Points of \( f \)**

Critical points occur where the derivative is either zero or undefined. Since the derivative \( f'(x) = (x + 4) e^{-x} \) is defined for all \( x \), we only need to find where \( f'(x) = 0 \).

\[
f'(x) = (x + 4) e^{-x} = 0
\]

Since \( e^{-x} \) is never zero, we set \( x + 4 = 0 \).

\[
x + 4 = 0 \implies x = -4
\]

Thus, the critical point is at \( x = -4 \).

### Part (b): **Intervals where \( f \) is Increasing or Decreasing**

To determine whether the function is increasing or decreasing, we analyze the sign of \( f'(x) \) on either side of the critical point \( x = -4 \).

- When \( x < -4 \), \( (x + 4) < 0 \) and \( e^{-x} > 0 \), so \( f'(x) < 0 \). Hence, \( f(x) \) is decreasing on \( (-\infty, -4) \).
- When \( x > -4 \), \( (x + 4) > 0 \) and \( e^{-x} > 0 \), so \( f'(x) > 0 \). Hence, \( f(x) \) is increasing on \( (-4, \infty) \).

Thus, \( f \) is decreasing on \( (-\infty, -4) \) and increasing on \( (-4, \infty) \).

### Part (c): **Local Maximum or Minimum Values**

At \( x = -4 \), the derivative changes sign from negative to positive. This indicates that \( f(x) \) has a **local minimum** at \( x = -4 \).

### Summary:
- (a) The critical point is at \( x = -4 \).
- (b) \( f(x) \) is decreasing on \( (-\infty, -4) \) and increasing on \( (-4, \infty) \).
- (c) There is a local minimum at \( x = -4 \).

Would you like more detailed steps or have any other questions? Here are some related questions for deeper understanding:

1. How would you determine the concavity of the function \( f(x) \)?
2. Can the second derivative test be applied to this function to confirm the local minimum?
3. How would a graph of \( f(x) \) look, based on the information we've derived?
4. What happens to the behavior of the function as \( x \to \infty \)?
5. How would the analysis change if the derivative were \( f'(x) = (x + a)e^{-x} \) for a general constant \( a \)?

**Tip**: For finding critical points, always consider both where the derivative equals zero and where it is undefined!

Answer the following questions about the function whose derivative is f'(x) = (x + 4)e^(-x).

Explore how to find the critical points, increasing/decreasing intervals, and local extrema of the function whose derivative is f'(x) = (x + 4)e^(-x). Suitable for high school students studying calculus.

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