Đáp án

\( (f'(x)f(x))' = (f'(x))^2 + f''(x)f(x) = 15x^4 + 12x \).

\( \implies  \int (f'(x)f(x))' dx = 3x^5 + 6x^2 + C \).

\( \implies f'(x)f(x) = 3x^5 + 6x^2 + C \).

\( f(0) = f'(0) = 1 \Rightarrow C = 1 \Rightarrow f'(x)f(x) = 3x^5 + 6x^2 + 1 \)

\( \Rightarrow \frac{(f(x))^2}{2} = \frac{x^6}{2} + 2x^3 + x + C \)

\( f(0) = 1 \Rightarrow C = \frac{1}{2} \)

\( (f(x))^2 = x^6 + 4x^3 + 2x + 1 \Rightarrow (f(1))^2 = 8 \Rightarrow f(1) = 2 \Rightarrow \boxed{D} \)