if p,q,r be real then the interval in which f(x) `=|underset(pr" "qr" "x+r^(2))underset(pq" "x+q^(2)" "qr)(x+p^(2)" "pq" "pr)|,`

Options

A. increase is `x lt -(2)/(3) (p^(2) +q^(2)+r^(2)), x gt0`
B. decrease is `(-(2)/(3)(p^(2) +q^(2)+r^(2)),0)`
C. decrease is `x lt - (2)/(3) (p^(2) +q^(2)+r^(2)) x gt 0`
D. increase is `(-(2)/(3) (p^(2)+q^(2)+r^(2)),0)`

Answer

To solve the problem, we need to analyze the function given by:

\[ f(x) = \left| \frac{(px + r^2)(qx + r^2)(x + p^2)}{(pq x + q^2)(qr x + r^2)(x + p^2)} \right| \]

### Step 1: Simplify the Function
First, we need to simplify the expression inside the absolute value. We can denote the numerator and denominator separately:

- **Numerator**: \( N(x) = (px + r^2)(qx + r^2)(x + p^2) \)
- **Denominator**: \( D(x) = (pq x + q^2)(qr x + r^2)(x + p^2) \)

### Step 2: Calculate the Derivative
Next, we need to find the derivative \( f'(x) \) to determine the intervals where the function is increasing or decreasing.

Using the quotient rule for derivatives, we have:

\[ f'(x) = \frac{N'(x)D(x) - N(x)D'(x)}{(D(x))^2} \]

### Step 3: Identify Critical Points
To find the critical points, we set \( f'(x) = 0 \). This occurs when:

\[ N'(x)D(x) - N(x)D'(x) = 0 \]

### Step 4: Analyze the Sign of the Derivative
We need to analyze the sign of \( f'(x) \) in the intervals determined by the critical points.

1. Find the roots of \( N'(x) = 0 \) and \( D'(x) = 0 \).
2. Determine the intervals based on these roots.

### Step 5: Determine Increasing and Decreasing Intervals
- The function \( f(x) \) is increasing where \( f'(x) > 0 \).
- The function \( f(x) \) is decreasing where \( f'(x) < 0 \).

### Step 6: Conclusion
Based on the analysis of the sign of the derivative, we can conclude the intervals for increasing and decreasing behavior.

### Final Result
The function \( f(x) \) is:
- **Increasing** in the intervals \( (-\infty, -\frac{2}{3}(p^2 + q^2 + r^2)) \cup (0, \infty) \)
- **Decreasing** in the interval \( (-\frac{2}{3}(p^2 + q^2 + r^2), 0) \)