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!

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.