1. **Problem statement:** We are given a cubic polynomial function $f(x) = ax^3 + bx^2 + cx + d$ with the following conditions: - The graph has a high point (local maximum) at $H(1|4)$. - $x_N = -1$ is a root of $f$, so $f(-1) = 0$. - $x_W = 3$ is an inflection point (Wendestelle) of $f$. 2. **General form and derivatives:** $$f(x) = ax^3 + bx^2 + cx + d$$ $$f'(x) = 3ax^2 + 2bx + c$$ $$f''(x) = 6ax + 2b$$ 3. **Using the given conditions to form equations:** - Since $x_N = -1$ is a root: $$f(-1) = a(-1)^3 + b(-1)^2 + c(-1) + d = -a + b - c + d = 0$$ - Since $H(1|4)$ is a high point, the function value and derivative at $x=1$ are: $$f(1) = a + b + c + d = 4$$ $$f'(1) = 3a(1)^2 + 2b(1) + c = 3a + 2b + c = 0$$ - Since $x_W = 3$ is an inflection point, the second derivative is zero there: $$f''(3) = 6a(3) + 2b = 18a + 2b = 0$$ 4. **Linear system of equations:** \[ \begin{cases} -a + b - c + d = 0 \\ a + b + c + d = 4 \\ 3a + 2b + c = 0 \\ 18a + 2b = 0 \end{cases} \] 5. **Summary:** This system can be solved to find $a$, $b$, $c$, and $d$. **Final answer:** The linear system describing the cubic polynomial with the given properties is: $$\begin{cases} -a + b - c + d = 0 \\ a + b + c + d = 4 \\ 3a + 2b + c = 0 \\ 18a + 2b = 0 \end{cases}$$