Cho hàm số bậc ba \[f\left( x \right) = {x^3} + a{x^2} + bx + c\,\,\,\left( {a,\,\,b,\,\,c \in \mathbb{R}} \right)\] thỏa mãn: \[f\left( 1 \right) = 10,f\left( 2 \right) = 20.\]. Khi đó \(\int\limits_0^3 {f'\left( x \right)dx} \) bằng:

Theo đề bài ta có

\(\begin{array}{l}\left\{ {\begin{array}{*{20}{c}}{f(1) = 10}\\{f(2) = 20}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{1 + a + b + c = 10}\\{{2^3} + {2^2}.a + 2b + c = 20}\end{array}} \right.\\ \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{a + b + c = 9}\\{4a + 2b + c = 12}\end{array}} \right. \Rightarrow 3a + b = 3\end{array}\)

\( \Rightarrow \int\limits_0^3 {f'\left( x \right)dx} = f(x)\left| {_0^3} \right. = f(3) - f(0)\)

\[ = {3^3} + {3^2}.a + 3b + c - c = 27 + 9a + 3b\]

\[ = 27 + 3(3a + b) = 27 + 3.3 = 36\]

Đáp án cần chọn là: D