Đáp án:

Xét: \( \int_0^1 \left( f'(x) \right)^2 dx - 6 \int_0^1 x^2 f'(x) dx + \int_0^1 9x^4 dx. \)

\( = \frac{9}{5} - \frac{18}{5}+\frac{9x^5}{5}|_0^1 =0 \)

\( \implies \int_0^1 \left( f'(x) - 3x^2 \right)^2 dx = 0 \implies f'(x) = 3x^2 \implies f(x) = x^3 + C. \)

\( f(1) = 0 \implies C = \): 
\( I = \int_0^1 x^3 dx =  \frac{x^4}{4}|_0^1 =  \frac{1}{4} \implies \boxed{B}\)

Giảng: Tìm \( k \) để: 
\( \int_0^1 \left( f'(x) - kx^2 \right)^2 dx = 0 \Leftrightarrow \int_0^1 \left( f'(x) \right)^2 dx - 2k \int_0^1 x^2 f'(x) dx + k^2 \int_0^1 x^4 dx = 0. \)

\( \frac{9}{5} - 2k  \frac{3}{5} +  \frac{k^2}{5} = 0 \Leftrightarrow  k^2 - 6k + 9 = 0 \Leftrightarrow k = 3. \)