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To solve this problem, let’s follow the steps needed to analyze the function \( f(x) = (3x + 1)e^x \):

1. **Find the critical numbers of \( f \) and \( f' \)**:
   - A critical point occurs where \( f'(x) = 0 \) or \( f'(x) \) is undefined. 
   
2. **Calculate \( f'(x) \)**:
   - Use the product rule: if \( f(x) = u(x)v(x) \) with \( u(x) = 3x + 1 \) and \( v(x) = e^x \), then
     \[
     f'(x) = u'(x)v(x) + u(x)v'(x).
     \]
   - Compute \( u'(x) = 3 \) and \( v'(x) = e^x \), so:
     \[
     f'(x) = 3e^x + (3x + 1)e^x = e^x(3x + 4).
     \]

3. **Solve \( f'(x) = 0 \)**:
   - Set \( e^x(3x + 4) = 0 \).
   - Since \( e^x \neq 0 \), we only need to solve \( 3x + 4 = 0 \):
     \[
     x = -\frac{4}{3}.
     \]
   - This is one of the critical points.

4. **Determine Concavity and Inflection Points**:
   - To analyze concavity, calculate \( f''(x) \).
   - Differentiate \( f'(x) = e^x(3x + 4) \) again using the product rule:
     \[
     f''(x) = e^x(3x + 4) + e^x \cdot 3 = e^x(3x + 7).
     \]
   - Set \( f''(x) = 0 \) to find potential inflection points:
     \[
     e^x(3x + 7) = 0.
     \]
   - Solving \( 3x + 7 = 0 \):
     \[
     x = -\frac{7}{3}.
     \]
   - So, \( x = -\frac{7}{3} \) is a potential inflection point.

5. **Behavior Analysis**:
   - With the critical points \( a = -\frac{4}{3} \) and \( b = -\frac{7}{3} \), analyze each interval:
     - **\( x < a \)**: Decreasing and concave downward.
     - **\( a < x < b \)**: Increasing and concave downward.
     - **\( x > b \)**: Increasing and concave upward.
  
6. **Limit Behavior at \( \pm \infty \)**:
   - As \( x \to -\infty \), \( f(x) \to 0 \).
   - As \( x \to \infty \), \( f(x) \to \infty \).

These steps should allow you to fill in the table:

- \( a = -\frac{4}{3} \), \( b = -\frac{7}{3} \)
- Behavior at each interval:
  - \( x < a \): Decreasing, concave downward.
  - \( a < x < b \): Increasing, concave downward.
  - \( x > b \): Increasing, concave upward.
- At \( -\infty \) and \( +\infty \): "Other"

Find all critical numbers of f and f' for f(x) = (3x + 1)e^x and analyze the behavior of f on different intervals.

Learn how to find the critical points of f(x) = (3x + 1)e^x and analyze its behavior over different intervals. This problem covers calculus concepts like critical points, concavity, inflection points, and limit behavior. Suitable for college-level calculus students.

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