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The task provided is a mathematical problem that deals with a family of functions \( f_a(x) \), where:

\[
f_a(x) = -\frac{1}{8}(a^2x^3 - 12ax^2 + 36x)
\]
with \( a \in \mathbb{R}^+ \) and \( x \in \mathbb{R} \).

Here’s a breakdown of what needs to be done:

1. **Symmetry**: You need to examine the function \( f_a(x) \) for symmetry with respect to the y-axis and the origin.
   
2. **Zeros (Nullstellen)**: Find the zeros of the function \( f_a(x) \), i.e., solve \( f_a(x) = 0 \).

3. **Extremum Points**: Determine the location and type (minimum or maximum) of the extremum points of the function \( f_a(x) \).

4. **Graph of \( G_2 \)**: Sketch the graph of \( G_2 \), where \( a = 2 \).

I will go step by step to address each part of the problem.

---

### Step 1: Symmetry
The function \( f_a(x) \) has cubic and quadratic terms, so we must check for symmetry:
- **y-axis symmetry**: A function is symmetric with respect to the y-axis if \( f(-x) = f(x) \). This usually applies to even functions.
- **Origin symmetry**: A function is symmetric with respect to the origin if \( f(-x) = -f(x) \), which applies to odd functions.

### Step 2: Zeros (Nullstellen)
To find the zeros of \( f_a(x) \), set the function equal to zero:
\[
f_a(x) = -\frac{1}{8}(a^2x^3 - 12ax^2 + 36x) = 0
\]
Solve for \( x \).

### Step 3: Extremum Points
To find the extremum points, we need to calculate the derivative \( f'_a(x) \) and solve \( f'_a(x) = 0 \) to locate the critical points. Then, use the second derivative \( f''_a(x) \) to classify these points as minima or maxima.

### Step 4: Graph of \( G_2 \)
Once we have the zeros and extremum points, we will sketch the graph of \( f_a(x) \) for \( a = 2 \).

---

I’ll begin with these steps to find the zeros and critical points of \( f_a(x) \).

Let me start by calculating the derivative of \( f_a(x) \) and solving for the extremum points.The first derivative of the function \( f_a(x) \) is:

\[
f'_a(x) = -0.375 a^2 x^2 + 3.0 a x - 4.5
\]

Setting \( f'_a(x) = 0 \) yields the critical points:

\[
x = \frac{2}{a} \quad \text{and} \quad x = \frac{6}{a}
\]

These are the locations of the extremum points of the function.

Next, I will check the second derivative to classify these points as minima or maxima, and proceed to find the zeros (roots) of \( f_a(x) \).It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can still provide you with some theoretical insights if you'd like to continue manually solving the problem. Here’s what we found so far:

1. **First derivative**: The critical points of the function are \( x = \frac{2}{a} \) and \( x = \frac{6}{a} \), which are potential minima or maxima.
2. To determine whether these points are minima or maxima, the second derivative needs to be evaluated at these points.
3. For the zeros (roots) of the function \( f_a(x) \), you will need to set the function equal to zero and solve for \( x \), leading to a cubic equation.

Would you like more details on the manual steps, or do you have any specific questions about the process? 

### Here are 5 questions that build on this information:
1. How would you use the second derivative test to classify the extremum points as minima or maxima?
2. Can you describe the process to find the zeros of a cubic function like \( f_a(x) \)?
3. What role does the parameter \( a \) play in shaping the graph of the function?
4. How would symmetry with respect to the y-axis or origin affect the sketch of the function \( G_2 \)?
5. Why is it important to find both the zeros and extremum points for sketching the graph?

**Tip**: When solving higher-degree polynomial equations, it’s often useful to factor the expression or use numerical methods for more complex roots.

Gegeben ist die Funktionenfamilie f_a(x): x → -1/8(a^2x^3 − 12ax^2 + 36x) mit a ∈ R^+ und x ∈ R. Bestimmen Sie die Nullstellen, untersuchen Sie auf Symmetrie zur y-Achse bzw. zum Ursprung, und finden Sie die Extrempunkte. Skizzieren Sie dann G_2.

Analyze the cubic function f_a(x) = -1/8(a^2x^3 − 12ax^2 + 36x), explore its roots (Nullstellen), symmetry, and extrema. Includes steps for derivative-based extrema analysis and graphing G_2 where a = 2.

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