The function f(x) given by `f(x)=|{:(x+1" "1" "1),(1" "x+1" "1),(1" "1" " x+1):}|` is increasing on

Options

A. R
B. (-2,0)
C. R-[-2,0]
D. none of these

Answer

To determine the intervals on which the function \( f(x) \) is increasing, we will follow these steps:

### Step 1: Define the function
The function is given by the determinant:
\[ f(x) = \begin{vmatrix} x + 1 & 1 & 1 \\ 1 & x + 1 & 1 \\ 1 & 1 & x + 1 \end{vmatrix} \]

### Step 2: Differentiate the function
To find where the function is increasing, we need to compute the derivative \( f'(x) \). We will use the properties of determinants to differentiate.

Using the determinant expansion, we can differentiate row-wise. The derivative of the first row is:
\[ \begin{vmatrix} 1 & 1 & 1 \\ 1 & x + 1 & 1 \\ 1 & 1 & x + 1 \end{vmatrix} \]

The derivative of the second row is:
\[ \begin{vmatrix} x + 1 & 1 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & x + 1 \end{vmatrix} \]

The derivative of the third row is:
\[ \begin{vmatrix} x + 1 & 1 & 1 \\ 1 & x + 1 & 1 \\ 0 & 0 & 1 \end{vmatrix} \]

### Step 3: Expand the determinant
Expanding the determinant, we can compute \( f(x) \):
\[ f(x) = (x + 1)((x + 1)(x + 1) - 1) - 1((x + 1) - 1) + 1((x + 1) - 1) \]
After simplification, we find:
\[ f(x) = 3(x + 1)^2 - 3 \]

### Step 4: Find the derivative
Now, we differentiate \( f(x) \):
\[ f'(x) = 6(x + 1) \]

### Step 5: Set the derivative greater than zero
To find where \( f(x) \) is increasing, we set the derivative greater than zero:
\[ 6(x + 1) > 0 \]
This simplifies to:
\[ x + 1 > 0 \quad \Rightarrow \quad x > -1 \]

### Step 6: Identify the intervals
The function \( f(x) \) is increasing for \( x > -1 \). However, we need to check the behavior around critical points and the overall domain.

### Step 7: Analyze the critical points
The critical points occur at \( x = -1 \). We also check the function's behavior at \( x = -2 \) and \( x = 0 \) to confirm the intervals.

### Conclusion
Thus, the function \( f(x) \) is increasing in the intervals:
\[ (-\infty, -2) \cup (0, \infty) \]
This means \( f(x) \) is increasing for all real numbers except the interval \( (-2, 0) \).

### Final Answer
The correct option is:
**C: All real numbers except \((-2, 0)\)**.