Đáp án:

\(\frac{(f'(x) (f(x))^2)^2}{1 + (f(x))^3} = \frac{1}{x+1}\)

\(\Rightarrow \frac{(3f'(x)f(x)^2)^2}{1 + (f(x))^3} = \frac{9}{x+1} \) (*)

Đặt \(h(x) = 1 + (f(x))^3 \Rightarrow  h'(x) = 3f'(x)(f(x))^2\)

(*) \(\Leftrightarrow \frac{(h'(x))^2}{h(x)} = \frac{9}{x+1} \Leftrightarrow  \frac{h'(x)}{\sqrt{h(x)}} =  \frac{3}{\sqrt{x+1}}\)

\(\Rightarrow \int \frac{h'(x)}{\sqrt{h(x)}} dx = \int \frac{3}{\sqrt{x+1}} dx \Rightarrow 2\sqrt{h(x)} = 6\sqrt{x+1} + C \quad \Rightarrow \quad h(x)= 9\sqrt{x+1} + C.\)

\(h(0) = 1 + (f(0))^3 = 9 \Rightarrow C = 0.\)

\(\Rightarrow h(x) = 9(x+1) \quad \Rightarrow \quad 1 + (f(x))^3 = 9x + 9.\)

\(\Rightarrow f(x) = \sqrt[3]{9x + 8}.\)

\(f(1) = \sqrt[3]{17} \approx 2.571 \quad \Rightarrow \boxed{\text{B}}.\)