This problem concerns the graph of a polynomial function of degree 4 and requires selecting the correct statements regarding the behavior of the graph in relation to the x-axis.

Analyzing the problem:

1. Polynomial properties:

   • Degree 4 polynomial functions generally have up to 4 roots (real or complex).
   • The graph shown has three critical points (maxima and minima), which indicates it has two points of inflection (where $f''(x) = 0$).

2. Graph interpretation:

   • The graph suggests possible roots (where $f(x) = 0$) and critical points.
   • Some roots may not cross the x-axis (e.g., if the x-axis lies below the graph).

3. Evaluating each statement:

   • First statement: "Es gibt genau zwei Stellen $x_1$ und $x_2$ mit $f(x_1) = 0$ und $f(x_2) = 0$."
     This suggests that there are exactly two roots. The graph does not show the x-axis, so we cannot definitively determine this based on the image. It is plausible but uncertain.

   • Second statement: "Es gibt genau zwei Stellen $x_1$ und $x_2$ mit $f''(x_1) = 0$ und $f''(x_2) = 0$."
     A degree-4 polynomial typically has two inflection points. Observing the graph's curvature, this statement is correct.

   • Third statement: "Es gibt genau eine Stelle $x_1$ mit $f''(x_1) = 0$."
     This contradicts the behavior of degree-4 polynomials, which generally have two inflection points. This statement is false.

   • Fourth statement: "Es gibt genau eine Stelle $x_1$ mit $f'(x_1) = 0$ und $f''(x_1) > 0$."
     This refers to a local minimum. The graph shows one local minimum with $f''(x_1) > 0$. This statement is correct.

   • Fifth statement: "Es gibt genau eine Stelle $x_1$ mit $f'(x_1) > 0$ und $f''(x_1) > 0$."
     This describes a region where the graph is increasing and concave up. This is not true, as no such point exists on this graph.

Correct answers:

• Second statement: $f''(x_1) = 0$ at two points.
• Fourth statement: $f'(x_1) = 0$ and $f''(x_1) > 0$ at a local minimum.

Let me know if you'd like a more detailed explanation or any follow-up!

Expansion Questions:

1. What are the general properties of a degree-4 polynomial?
2. How do inflection points relate to the second derivative?
3. Can you describe why a degree-4 polynomial typically has three critical points?
4. How does the graph of a polynomial behave as $x \to \pm\infty$?
5. Why are there usually exactly two inflection points in this type of polynomial?

Tip:

Always verify whether the x-axis placement impacts the interpretation of roots and extrema in problems like this!