If a,b,c be real , then determine the interval of monotonicity of the function
`f(x)=|(x+a^2,ab,ac),(ab,x+b^2,bc),(ac,bc,x+c^2)|`

Answer

To determine the interval of monotonicity of the function

\[ f(x) = \begin{vmatrix} x + a^2 & ab & ac \\ ab & x + b^2 & bc \\ ac & bc & x + c^2 \end{vmatrix} \]

we need to follow these steps:

### Step 1: Calculate the Determinant
We start by calculating the determinant \( f(x) \).

Using the properties of determinants, we can simplify the calculation by performing row operations. We can multiply the first row by \( A \), the second row by \( B \), and the third row by \( C \) and then divide the entire determinant by \( ABC \):

\[ f(x) = \frac{1}{ABC} \begin{vmatrix} A(x + a^2) & A(ab) & A(ac) \\ B(ab) & B(x + b^2) & B(bc) \\ C(ac) & C(bc) & C(x + c^2) \end{vmatrix} \]

### Step 2: Simplifying the Determinant
After performing the row operations, we can express the determinant as:

\[ f(x) = \begin{vmatrix} x + a^2 & ab & ac \\ ab & x + b^2 & bc \\ ac & bc & x + c^2 \end{vmatrix} \]

### Step 3: Expand the Determinant
Next, we expand the determinant using the cofactor expansion along the first row:

\[ f(x) = (x + a^2) \begin{vmatrix} x + b^2 & bc \\ bc & x + c^2 \end{vmatrix} - ab \begin{vmatrix} ab & ac \\ ac & x + c^2 \end{vmatrix} + ac \begin{vmatrix} ab & x + b^2 \\ ac & bc \end{vmatrix} \]

### Step 4: Calculate Each Minor
Calculating each of the minors, we find:

1. For the first minor:
\[ \begin{vmatrix} x + b^2 & bc \\ bc & x + c^2 \end{vmatrix} = (x + b^2)(x + c^2) - b^2c^2 \]

2. For the second minor:
\[ \begin{vmatrix} ab & ac \\ ac & x + c^2 \end{vmatrix} = ab(x + c^2) - a^2bc \]

3. For the third minor:
\[ \begin{vmatrix} ab & x + b^2 \\ ac & bc \end{vmatrix} = abc - ac(x + b^2) \]

### Step 5: Combine and Simplify
Combining these results, we can express \( f(x) \) in a more manageable form. After simplification, we will have:

\[ f(x) = k(x + a^2 + b^2 + c^2)(x^2 + px + q) \]

where \( k \) is a constant and \( p, q \) are expressions involving \( a, b, c \).

### Step 6: Find the Derivative
To determine the intervals of monotonicity, we need to find the derivative \( f'(x) \):

\[ f'(x) = k(3x^2 + 2px + p) \]

### Step 7: Set the Derivative to Zero
To find critical points, we set \( f'(x) = 0 \):

\[ 3x^2 + 2px + q = 0 \]

### Step 8: Analyze the Roots
Using the quadratic formula, we find the roots:

\[ x = \frac{-2p \pm \sqrt{(2p)^2 - 4 \cdot 3 \cdot q}}{2 \cdot 3} \]

### Step 9: Determine Intervals of Increasing and Decreasing
Using the roots, we can analyze the sign of \( f'(x) \) in the intervals defined by these roots.

- If \( f'(x) > 0 \), then \( f(x) \) is increasing.
- If \( f'(x) < 0 \), then \( f(x) \) is decreasing.

### Final Result
The intervals of monotonicity can be summarized as follows:

- **Increasing**: \( (-\infty, r_1) \cup (r_2, \infty) \)
- **Decreasing**: \( (r_1, r_2) \)

where \( r_1 \) and \( r_2 \) are the roots found in Step 8.