Question:

<p><strong>Step 1:</strong> Recognize that $f(x)$ is a polynomial of degree at most 3 in $x$. We need the coefficient of $x^1$.</p> <p><strong>Step 2:</strong> Use column operations to simplify. Subtract column 2 from column 1 and column 3:</p> <p>$C_1 \to C_1 - C_2$, $C_3 \to C_3 - C_2$</p> <p><strong>Step 3:</strong> After subtraction:</p> $$\begin{vmatrix} c_1 - b & x+a & c_1-c_2 \\ -a & x+c_2 & c_2-a \\ -a & x+b & c_3-a \end{vmatrix}$$ <p><strong>Step 4:</strong> The determinant has the form: (constant terms) $+$ (coefficient of $x$) $\cdot x$ $+$ higher order terms.</p> <p><strong>Step 5:</strong> Expanding along column 2 (which isolates the $x$ terms):</p> <p>$f(x) = (x+a) \cdot M_1 + (x+c_2) \cdot M_2 + (x+b) \cdot M_3$ where $M_i$ are $2 \times 2$ minors.</p> <p><strong>Step 6:</strong> The coefficient of $x$ equals the sum of all $2 \times 2$ minors obtained by removing row $i$ and column 2.</p> <p><strong>Step 7:</strong> By careful expansion (or using the relationship with $g(x)$): coefficient of $x$ in $f(x)$ is $$\frac{g(a)-g(b)}{b-a}$$</p> <p>$\therefore$ Answer: C</p>