Consider the function f(x) = `|{:(a^(2)+x,,ab,,ac),(ab,,b^(2)+x,,bc),(ac,,bc,,c^(2)+x):}|`
In which of the following interval f(x) is strictly increasing

Options

A. `(-oo ,oo)`
B. `(-oo ,0)`
C. `(0,oo)`
D. None of these

Explanation

To determine the interval in which the function \( f(x) = \begin{vmatrix} a^2 + x & ab & ac \\ ab & b^2 + x & bc \\ ac & bc & c^2 + x \end{vmatrix} \) is strictly increasing, we will first compute the determinant and then analyze its behavior with respect to \( x \).

### Step 1: Compute the Determinant

We start with the determinant:

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

### Step 2: Expand the Determinant

Using the properties of determinants, we can expand this determinant. We will use the first row for expansion:

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

### Step 3: Calculate the 2x2 Determinants

Now, we calculate the 2x2 determinants:

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

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

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

### Step 4: Substitute Back into the Determinant

Now substituting back into the determinant:

\[ \Delta = (a^2 + x)((b^2 + x)(c^2 + x) - b^2c^2) - ab(abx) + ac(ab^2c - acb^2 - acx) \]

### Step 5: Simplify the Expression

After simplifying the expression, we will have a polynomial in \( x \). The function \( f(x) \) will be strictly increasing where its derivative \( f'(x) \) is positive.

### Step 6: Find the Derivative

To find \( f'(x) \), we differentiate \( \Delta \) with respect to \( x \):

\[ f'(x) = \frac{d}{dx} \Delta \]

### Step 7: Analyze the Sign of the Derivative

We need to find the intervals where \( f'(x) > 0 \). This will involve solving the inequality derived from the expression of \( f'(x) \).

### Step 8: Identify the Interval

After solving the inequality, we will determine the interval(s) where \( f(x) \) is strictly increasing.

### Conclusion

Thus, the final answer will be the interval(s) where \( f(x) \) is strictly increasing based on the analysis of \( f'(x) \).

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