1. Problem 9: Given $$f(x) = (x-1)(x+2)\left(x-\frac{8}{7}-\frac{1}{7}x\right), \quad g(x) = (x+7)(x+8)(x-1) - (x-7)(x+2)(x-12), \quad \log f^{(x)} = 2 \log g^{(x)}$$ We want to analyze the relation between $f$ and $g$. 2. Simplify $f(x)$: $$f(x) = (x-1)(x+2)\left(x - \frac{8}{7} - \frac{1}{7}x\right) = (x-1)(x+2)\left(\frac{7x - 8 - x}{7}\right) = (x-1)(x+2)\frac{6x - 8}{7}$$ 3. Simplify $g(x)$: Expand each product: $$(x+7)(x+8)(x-1) = (x^2 + 15x + 56)(x-1) = x^3 - x^2 + 15x^2 - 15x + 56x - 56 = x^3 + 14x^2 + 41x - 56$$ $$(x-7)(x+2)(x-12) = (x^2 - 5x - 14)(x-12) = x^3 - 12x^2 - 5x^2 + 60x - 14x + 168 = x^3 - 17x^2 + 46x + 168$$ 4. Compute $g(x)$: $$g(x) = (x^3 + 14x^2 + 41x - 56) - (x^3 - 17x^2 + 46x + 168) = 0 + 31x^2 - 5x - 224$$ 5. Given $\log f^{(x)} = 2 \log g^{(x)}$, this implies: $$\log f^{(x)} = \log (g^{(x)})^2 \Rightarrow f^{(x)} = (g^{(x)})^2$$ 6. Problem 10: Given $$f(x) = \frac{f'(x) + x}{4}, \quad a=1, b=6, c=2, d=3$$ Rewrite: $$4f(x) = f'(x) + x \Rightarrow f'(x) - 4f(x) = -x$$ 7. Solve the linear ODE: Homogeneous solution: $$f'_h - 4f_h = 0 \Rightarrow f_h = Ce^{4x}$$ Particular solution guess: $$f_p = Ax + B$$ Substitute: $$f'_p = A$$ $$A - 4(Ax + B) = -x \Rightarrow A - 4Ax - 4B = -x$$ Match coefficients: $$-4A = -1 \Rightarrow A = \frac{1}{4}$$ $$A - 4B = 0 \Rightarrow \frac{1}{4} - 4B = 0 \Rightarrow B = \frac{1}{16}$$ 8. General solution: $$f(x) = Ce^{4x} + \frac{1}{4}x + \frac{1}{16}$$ 9. Problem 11: Given $$\log f = 1 + \frac{\sqrt{2\pi x}}{x}$$ This is a function definition; no equation to solve. 10. Problem 12: Given $$f(x) = (x+3)(x+4)$$ Find tangential vector value (derivative): $$f(x) = x^2 + 7x + 12$$ $$f'(x) = 2x + 7$$ 11. Problem 13: Given $$\log f = 1 - x$$ Rewrite: $$f = e^{1-x} = e \cdot e^{-x}$$ Derivative: $$f' = -e^{1-x} = -f$$ 12. Problem 15: Given $$\log f = (x^3 - 2) + 4 = x^3 + 2$$ Rewrite: $$f = e^{x^3 + 2}$$ Derivative: $$f' = 3x^2 e^{x^3 + 2} = 3x^2 f$$ 13. Problem 16: Given $$\log f = (x^3 - 26.5x) - x = x^3 - 27.5x$$ Rewrite: $$f = e^{x^3 - 27.5x}$$ Derivative: $$f' = (3x^2 - 27.5) e^{x^3 - 27.5x} = (3x^2 - 27.5) f$$ 14. Problem 17: Given $$x_0 = \frac{a}{b} + \frac{c}{d}$$ Substitute values from problem 10: $$x_0 = \frac{1}{6} + \frac{2}{3} = \frac{1}{6} + \frac{4}{6} = \frac{5}{6}$$ 15. Problem 27: Compute $$(0.5 - 11.78)^2 = (-11.28)^2 = 127.2384$$ 16. Problem 23: Given $$\log f = \frac{x}{(x-1)^4 + 1}$$ Rewrite: $$f = e^{\frac{x}{(x-1)^4 + 1}}$$ Derivative using quotient and chain rule is complex; no explicit request to solve. 17. Problem 25: Given $$\log f = x - 2$$ Rewrite: $$f = e^{x-2}$$ Derivative: $$f' = e^{x-2} = f$$ 18. Problem 26: Given $$\log f = x^2 + \sqrt{x+2}$$ Rewrite: $$f = e^{x^2 + \sqrt{x+2}}$$ Derivative: $$f' = \left(2x + \frac{1}{2\sqrt{x+2}}\right) f$$ 19. Problem 27 (again): Given $$\log f = x - x^2$$ Rewrite: $$f = e^{x - x^2}$$ Derivative: $$f' = (1 - 2x) e^{x - x^2} = (1 - 2x) f$$ 20. Given relation: $$\frac{f'}{f} = \frac{g}{f} + \frac{g'}{g} \frac{g}{f}$$ and $$\frac{d}{dx} \log f = 0$$ Implies $\log f$ is constant. 21. Summary: Each problem was solved or simplified stepwise.