Let `f(x)=|{:(p+x,q+x,r+x),(l+x,m+x,n+x),(a+x,b+x,c+x):}|` then

Options

A. `f''(x)=0`
B. f''(x) is a second degree function of x
C. f'(x) is a linear function of x
D. none of these

Answer

To solve the problem involving the function \( f(x) = |(p+x, q+x, r+x), (l+x, m+x, n+x), (a+x, b+x, c+x)| \), we need to analyze the determinant formed by the given vectors. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Determinant
The function \( f(x) \) is defined as the determinant of a 3x3 matrix formed by the vectors:
\[ \begin{pmatrix} p+x & q+x & r+x \\ l+x & m+x & n+x \\ a+x & b+x & c+x \end{pmatrix} \]
This determinant will depend on the variable \( x \).

### Step 2: Expand the Determinant
To find \( f(x) \), we will expand the determinant using the properties of determinants. The determinant of a 3x3 matrix can be calculated using:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is represented as:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

### Step 3: Calculate the Determinant
For our specific matrix, we can calculate the determinant as follows:
\[ f(x) = (p+x) \begin{vmatrix} m+x & n+x \\ b+x & c+x \end{vmatrix} - (q+x) \begin{vmatrix} l+x & n+x \\ a+x & c+x \end{vmatrix} + (r+x) \begin{vmatrix} l+x & m+x \\ a+x & b+x \end{vmatrix} \]

### Step 4: Simplify Each Minor
Each of the minors can be simplified:
1. \( \begin{vmatrix} m+x & n+x \\ b+x & c+x \end{vmatrix} = (m+x)(c+x) - (n+x)(b+x) \)
2. \( \begin{vmatrix} l+x & n+x \\ a+x & c+x \end{vmatrix} = (l+x)(c+x) - (n+x)(a+x) \)
3. \( \begin{vmatrix} l+x & m+x \\ a+x & b+x \end{vmatrix} = (l+x)(b+x) - (m+x)(a+x) \)

### Step 5: Combine and Collect Terms
After substituting back into the determinant expression, we will collect all terms involving \( x \). The highest power of \( x \) will determine the nature of \( f(x) \).

### Step 6: Determine the Degree of \( f(x) \)
After simplification, we will find that the highest power of \( x \) in \( f(x) \) is 1. Thus, we can express \( f(x) \) in the form:
\[ f(x) = ax + b \]
where \( a \) and \( b \) are constants determined by the coefficients of the linear terms.

### Step 7: Find the Derivatives
1. The first derivative \( f'(x) \) will be:
\[ f'(x) = a \]
which is a constant.
2. The second derivative \( f''(x) \) will be:
\[ f''(x) = 0 \]

### Conclusion
Since \( f''(x) = 0 \), the function \( f(x) \) is linear in \( x \), confirming that the second derivative is constant.