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To solve this problem, let's go through the necessary steps to find the critical points and behaviors of the function \( f(x) = (3x + 1)e^x \).

### Step 1: Find the critical numbers
1. **Find \( f'(x) \)**: Using the product rule, since \( f(x) = u \cdot v \) where \( u = 3x + 1 \) and \( v = e^x \):
   \[
   f'(x) = u'v + uv' = 3e^x + (3x + 1)e^x = e^x(3 + 3x + 1) = e^x(3x + 4)
   \]
   
2. **Set \( f'(x) = 0 \)**: To find critical points, set \( f'(x) = 0 \).
   \[
   e^x(3x + 4) = 0
   \]
   Since \( e^x \neq 0 \), we solve \( 3x + 4 = 0 \):
   \[
   x = -\frac{4}{3}
   \]
   
   So, \( x = -\frac{4}{3} \) is a critical point.

### Step 2: Determine behavior at critical points and infinity
Now, we can fill in the table with critical points and behaviors:

1. **At \( x = -\frac{4}{3} \)** (let \( a = -\frac{4}{3} \))
2. Choose the other interval values accordingly based on \( x < a \), \( a < x < b \), and \( x > b \).

Let me know if you’d like detailed steps on filling each behavior in the dropdowns or have any questions!

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**Additional Practice Questions:**
1. What is the significance of critical points in determining the shape of a graph?
2. How would the graph change if the coefficient in \( 3x + 1 \) was different?
3. How does the exponential function \( e^x \) influence the behavior of \( f(x) \) as \( x \to \infty \)?
4. What would happen to the critical point if the function was \( (2x + 3)e^x \) instead?
5. How can we determine concavity for the function \( f(x) \)?

**Tip:** In problems involving exponential functions, always remember that \( e^x \) is never zero, which simplifies finding critical points.

Analyze the function f(x) = (3x + 1)e^x, find its critical numbers and behaviors over different intervals.

This problem involves finding the critical points of the function f(x) = (3x + 1)e^x by deriving and analyzing its behavior over various intervals. Explore how to apply the product rule to differentiate, set up the equation f'(x) = 0, and understand the behavior of the function near critical points.

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