If `(-3, -1)` is the largest interval in which the function `f(x)=x^(3)+6x^(2)+ax+2` is decreasing, then `[a]` is equal to (where, `[.]` denotes the greatest integer function)

Options

A. 8
B. 9
C. 10
D. 11

Answer

To solve the problem, we need to determine the value of \( a \) such that the function \( f(x) = x^3 + 6x^2 + ax + 2 \) is decreasing in the interval \( (-3, -1) \).

### Step-by-Step Solution:

1. **Find the derivative of the function**:
The first step is to find the derivative of \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^3 + 6x^2 + ax + 2) = 3x^2 + 12x + a \]

2. **Set the derivative less than or equal to zero**:
Since the function is decreasing in the interval \( (-3, -1) \), we need:
\[ f'(x) \leq 0 \quad \text{for } x \in (-3, -1) \]

3. **Identify the critical points**:
The critical points occur when \( f'(x) = 0 \):
\[ 3x^2 + 12x + a = 0 \]
The roots of this quadratic equation will be the points where the function changes from increasing to decreasing or vice versa.

4. **Use the roots of the derivative**:
We know that the function is decreasing between the roots, which are given as \( -3 \) and \( -1 \). According to Vieta's formulas, the sum of the roots \( (-3) + (-1) = -4 \) and the product of the roots \( (-3)(-1) = 3 \) can be used to find \( a \).

5. **Relate the coefficients to the roots**:
From Vieta's formulas:
- The sum of the roots \( -\frac{b}{a} = -\frac{12}{3} = -4 \) (which is satisfied).
- The product of the roots \( \frac{c}{a} = \frac{a}{3} = 3 \).

Therefore, we can set up the equation:
\[ \frac{a}{3} = 3 \]

6. **Solve for \( a \)**:
Multiplying both sides by 3 gives:
\[ a = 9 \]

7. **Find the greatest integer function**:
The problem asks for \( [a] \), which is the greatest integer less than or equal to \( a \):
\[ [a] = [9] = 9 \]

### Final Answer:
Thus, the greatest integer of \( a \) is \( 9 \).