If a, b, c are real numbers, then find the intervals in which :
`f(x)=|(x+a^(2),ab,ac),(ab,x+b^(2),bc),(ac,bc,x+c^(2))|` is strictly increasing or decreasing.

Answer

To find the intervals in which the function \( f(x) = \left| \begin{array}{ccc} x + a^2 & ab & ac \\ ab & x + b^2 & bc \\ ac & bc & x + c^2 \end{array} \right| \) is strictly increasing or decreasing, we will follow these steps:

### Step 1: Differentiate the function
To determine where the function is increasing or decreasing, we first need to find the derivative \( f'(x) \). The determinant can be differentiated using the properties of determinants.

### Step 2: Calculate the determinant
Let's denote the determinant as \( D \):
\[ D = \left| \begin{array}{ccc} x + a^2 & ab & ac \\ ab & x + b^2 & bc \\ ac & bc & x + c^2 \end{array} \right| \]

### Step 3: Differentiate the determinant
Using the differentiation of determinants, we differentiate each row with respect to \( x \):
1. Differentiate the first row:
\[ \frac{d}{dx}(x + a^2) = 1, \quad \frac{d}{dx}(ab) = 0, \quad \frac{d}{dx}(ac) = 0 \]
The first row becomes \( (1, 0, 0) \).

2. Differentiate the second row:
\[ \frac{d}{dx}(ab) = 0, \quad \frac{d}{dx}(x + b^2) = 1, \quad \frac{d}{dx}(bc) = 0 \]
The second row becomes \( (0, 1, 0) \).

3. Differentiate the third row:
\[ \frac{d}{dx}(ac) = 0, \quad \frac{d}{dx}(bc) = 0, \quad \frac{d}{dx}(x + c^2) = 1 \]
The third row becomes \( (0, 0, 1) \).

### Step 4: Calculate the new determinant
Now, we compute the determinant of the matrix formed by these derivatives:
\[ f'(x) = \left| \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right| = 1 \]

### Step 5: Analyze the sign of \( f'(x) \)
Since \( f'(x) = 1 \), which is positive for all \( x \), we conclude that:
- The function \( f(x) \) is strictly increasing for all \( x \).

### Step 6: Conclusion
Thus, the intervals in which \( f(x) \) is strictly increasing or decreasing are:
- **Strictly Increasing**: \( (-\infty, \infty) \)
- **Strictly Decreasing**: There are no intervals where \( f(x) \) is strictly decreasing.