`f(x) = |{:(x+c_(1),,x+a,,x+a),(x+b,,x+c_(2),,x+a),(x+b,,x+b,,x+c_(3)):}| " and " g(x)= (C_(1) -x)(c_(3)-x)`
Coefficient of x in f(x) is

Options

A. `(g(a)-f(b))/(b-a)`
B. `(g(-a)-g(-b))/(b-a)`
C. `(g(a)-g(b))/(b-a)`
D. none of these

Explanation

To find the coefficient of \( x \) in the determinant function \( f(x) \), we will follow these steps:

### Step 1: Write the Determinant
The function \( f(x) \) is given as:
\[ f(x) = \begin{vmatrix} x + c_1 & x + b & x + b \\ x + c_2 & x + a & x + b \\ x + c_3 & x + a & x + a \end{vmatrix} \]

### Step 2: Apply Column Operations
We can simplify the determinant by performing column operations. We will subtract the first column from the second and third columns:
\[ f(x) = \begin{vmatrix} x + c_1 & (x + b) - (x + c_1) & (x + b) - (x + c_1) \\ x + c_2 & (x + a) - (x + c_2) & (x + b) - (x + c_2) \\ x + c_3 & (x + a) - (x + c_3) & (x + a) - (x + c_3) \end{vmatrix} \]
This simplifies to:
\[ f(x) = \begin{vmatrix} x + c_1 & b - c_1 & b - c_1 \\ x + c_2 & a - c_2 & b - c_2 \\ x + c_3 & a - c_3 & a - c_3 \end{vmatrix} \]

### Step 3: Factor Out Common Terms
Now, we can factor out \( x \) from the first column:
\[ f(x) = \begin{vmatrix} x & 1 & 1 \\ x & 1 & 1 \\ x & 1 & 1 \end{vmatrix} + \text{(terms without x)} \]
The first column now has \( x \) factored out, and we can compute the determinant of the remaining matrix.

### Step 4: Calculate the Determinant
The determinant will be:
\[ f(x) = x \cdot \begin{vmatrix} 1 & b - c_1 & b - c_1 \\ 1 & a - c_2 & b - c_2 \\ 1 & a - c_3 & a - c_3 \end{vmatrix} \]
Now, we can compute the determinant of the \( 2 \times 2 \) matrix.

### Step 5: Expand the Determinant
The determinant simplifies to:
\[ = (b - c_1)(a - c_3) - (b - c_1)(a - c_2) + \text{(other terms)} \]
The coefficient of \( x \) will be the sum of the products of the constants from the determinant.

### Step 6: Identify the Coefficient of \( x \)
The coefficient of \( x \) in \( f(x) \) will be the result of the determinant evaluated at \( x = 0 \).

### Final Answer
The coefficient of \( x \) in \( f(x) \) is:
\[ \text{Coefficient of } x = (b - c_1)(a - c_3) - (b - c_1)(a - c_2) \]