1. The problem asks to analyze the functions given in problems 12 and 13, focusing on finding inflection points (Wendepunkte) and other characteristics. 2. For problem 12, the function is $f(x) = \frac{1}{3}x^3 - x^2$. 3. The first derivative is $f'(x) = x^2 - 2x$. 4. The second derivative is $f''(x) = 2x - 2$. 5. To find inflection points, set $f''(x) = 0$: $$2x - 2 = 0$$ Add 2 to both sides: $$2x = 2$$ Divide both sides by 2: $$x = 1$$ 6. Evaluate $f(x)$ at $x=1$: $$f(1) = \frac{1}{3}(1)^3 - (1)^2 = \frac{1}{3} - 1 = -\frac{2}{3}$$ 7. The third derivative is $f'''(x) = 2$, which is nonzero, confirming an inflection point at $W(1|-\frac{2}{3})$. 8. For problem 13, the function is $f(x) = x^5 + 2$. 9. The first derivative is $f'(x) = 5x^4$. 10. The second derivative is $f''(x) = 20x^3$. 11. Set $f''(x) = 0$ to find inflection points: $$20x^3 = 0$$ Divide both sides by 20: $$\cancel{20}x^3 = 0 \Rightarrow x^3 = 0$$ 12. Solve for $x$: $$x = 0$$ 13. The third derivative is $f'''(x) = 60x^2$. 14. Evaluate $f'''(0)$: $$f'''(0) = 60 \times 0^2 = 0$$ 15. Since the third derivative at $x=0$ is zero, the graph has no inflection points. Final answers: - Problem 12: Inflection point at $W(1|-\frac{2}{3})$. - Problem 13: No inflection points.