The domain of `f(x)=1n(a x^3+(a+b)x^2+(b+c)x+c),` where `a >0,b^2-4a c=0,i s(w h e r e[dot]` represetns greatest integer function). `(-1,oo)~(-b/(2a))` (1,`oo)~{-b/(2a)}` `(-1,1)~{-b/(2a)}` `non eoft h e s e`

Options

A. `(-1,oo)~{-(b)/(2a)}`
B. `(1,oo)~{-(b)/(2a)}`
C. `(-1,1)~{-(b)/(2a)}`
D. None of these

Explanation

To find the domain of the function \( f(x) = \ln(ax^3 + (a+b)x^2 + (b+c)x + c) \) given that \( a > 0 \) and \( b^2 - 4ac = 0 \), we need to ensure that the argument of the logarithm is greater than zero.

### Step-by-Step Solution:

1. **Identify the condition for the logarithm:**
The natural logarithm \( \ln(x) \) is defined only for \( x > 0 \). Therefore, we need:
\[ ax^3 + (a+b)x^2 + (b+c)x + c > 0 \]

2. **Analyze the polynomial:**
The polynomial \( ax^3 + (a+b)x^2 + (b+c)x + c \) is a cubic polynomial. Since \( a > 0 \), the leading coefficient is positive, which means the polynomial will tend to \( +\infty \) as \( x \to +\infty \) and \( -\infty \) as \( x \to -\infty \).

3. **Find the roots of the polynomial:**
Given \( b^2 - 4ac = 0 \), this indicates that the quadratic part of the polynomial has a double root. We can find the double root using the quadratic formula:
\[ x = \frac{-b}{2a} \]
This means that the polynomial can be factored as:
\[ a(x + \frac{b}{2a})^2 \]
Hence, the polynomial can be rewritten as:
\[ a(x + \frac{b}{2a})^2 + \text{(linear term)} \]

4. **Determine the sign of the polynomial:**
The quadratic term \( (x + \frac{b}{2a})^2 \) is always non-negative and equals zero at \( x = -\frac{b}{2a} \). The linear term will determine the overall sign of the polynomial.

5. **Set the conditions for the domain:**
The polynomial is greater than zero for all \( x \) except at the double root \( x = -\frac{b}{2a} \). Therefore, we need to exclude this point from our domain.

6. **Combine the conditions:**
Since the polynomial is a cubic function with a double root, the domain of \( f(x) \) is:
\[ x \in (-1, \infty) \setminus \left\{-\frac{b}{2a}\right\} \]

### Final Domain:
Thus, the domain of the function \( f(x) \) is:
\[ (-1, \infty) \setminus \left\{-\frac{b}{2a}\right\} \]