if a,b,c are real then find the intervial in which `f(x)=|{:(x+a^2,ab,ac),(ab,x+b^2,bc),(ac,bc,x+c^2):}|` is decreasing.

Answer

To solve the problem of finding the interval 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 decreasing, we will follow these steps:

### Step 1: Differentiate the Determinant
To determine where \( f(x) \) is decreasing, we need to find the derivative \( f'(x) \) and set it less than 0. The determinant can be differentiated with respect to \( x \) using the properties of determinants.

### Step 2: Calculate \( f'(x) \)
Using the rule for differentiating determinants, we differentiate each row one by one:

1. **Differentiate the first row**:
\[ \frac{\partial}{\partial x} \left| \begin{array}{ccc} x + a^2 & ab & ac \\ ab & x + b^2 & bc \\ ac & bc & x + c^2 \end{array} \right| = \left| \begin{array}{ccc} 1 & 0 & 0 \\ ab & x + b^2 & bc \\ ac & bc & x + c^2 \end{array} \right| \]

2. **Differentiate the second row**:
\[ \left| \begin{array}{ccc} x + a^2 & ab & ac \\ 0 & 1 & 0 \\ ac & bc & x + c^2 \end{array} \right| \]

3. **Differentiate the third row**:
\[ \left| \begin{array}{ccc} x + a^2 & ab & ac \\ ab & x + b^2 & bc \\ 0 & 0 & 1 \end{array} \right| \]

### Step 3: Expand the Determinants
Now, we will expand these determinants. After performing the necessary calculations, we will arrive at:
\[ f'(x) = 3x^2 + 2x(a^2 + b^2 + c^2) \]

### Step 4: Set the Derivative Less Than Zero
To find the intervals where \( f(x) \) is decreasing, we need to solve:
\[ 3x^2 + 2x(a^2 + b^2 + c^2) < 0 \]

### Step 5: Factor the Expression
Factoring out \( x \):
\[ x(3x + 2(a^2 + b^2 + c^2)) < 0 \]

### Step 6: Find Critical Points
Setting the factors equal to zero gives us the critical points:
1. \( x = 0 \)
2. \( 3x + 2(a^2 + b^2 + c^2) = 0 \) leads to \( x = -\frac{2}{3}(a^2 + b^2 + c^2) \)

### Step 7: Determine the Intervals
The critical points divide the number line into intervals. We will test the sign of \( f'(x) \) in each interval:
1. \( (-\infty, -\frac{2}{3}(a^2 + b^2 + c^2)) \)
2. \( (-\frac{2}{3}(a^2 + b^2 + c^2), 0) \)
3. \( (0, \infty) \)

### Step 8: Analyze the Signs
- For \( x < -\frac{2}{3}(a^2 + b^2 + c^2) \), \( f'(x) > 0 \) (increasing).
- For \( -\frac{2}{3}(a^2 + b^2 + c^2) < x < 0 \), \( f'(x) < 0 \) (decreasing).
- For \( x > 0 \), \( f'(x) > 0 \) (increasing).

### Conclusion
Thus, the function \( f(x) \) is decreasing in the interval:
\[ \left(-\frac{2}{3}(a^2 + b^2 + c^2), 0\right) \]