Lời giải

• \( y = f(x) \), \( dy = \left(ax + \frac{b}{x^2}\right) dx \)    \( \Rightarrow f'(x) = ax + \frac{b}{x^2} \) 
 
\( f'(1) = 0 \Rightarrow a + b = 0 \)

• \( dy = \left(ax + \frac{b}{x^2}\right) dx \Rightarrow \int dy = \int \left(ax + \frac{b}{x^2}\right) dx \)   \( \Rightarrow y = \frac{a x^2}{2} - \frac{b}{x} + c = f(x) \)

• \( f(1) = 4 \Leftrightarrow \frac{a}{2} - b + c = 4 \)

• \( f(-1) = 2 \Leftrightarrow \frac{a}{2} + b + c = 2 \)

Vậy  \( \begin{cases}
a + b = 0 \\
\frac{a}{2} - b + c = 4 \\
\frac{a}{2} + b + c = 2
\end{cases} \Rightarrow
\begin{cases}
a = 1\\ 
b = -1\\
c = \frac{5}{2}
\end{cases}\)

 \( f(x) = \frac{ x^2}{2} + \frac{1}{x} + \frac{5}{2}  \Rightarrow f(-2) = 2 - \frac{1}{2} + \frac{5}{2} = 4 \)

\(\Rightarrow\) Vậy chọn đáp án \(\boxed{\text{B}} \)