Đáp án:

\( f(1) - f(0) = 3 \implies \int_0^1 f'(x) dx = 3. \)

Tìm \( k \) để: \( \int_0^1 \left[ f'(x) - k \right]^2 dx = C. \)

\( \Leftrightarrow \int_0^1 \left( f'(x) \right)^2 dx - 2k \int_0^1 f'(x) dx + k^2 \int_0^1 dx = 0. \)
\( \Leftrightarrow 9 - 6k + k^2 = 0 \implies  k = 3. \)

Suy ra: \( f'(x) = 3 \implies f(x) = 3x + C. \)
\( f(1) - f(0) = 3 \Leftrightarrow 3 - C + C = 3. \)

\( f(2) = 6  \implies C = 0 \)

\( \implies f(x) = 3x \implies  (f(x))^4 = 81x^4\)

\( I =  \int_0^1(f(x) x)^4 dx =   \frac{81x^5}{5} \big|_0^1 = \frac{81}{5} \implies \boxed{C} \)