Let `f (x) = x^3 +6x^2 + px + 2`. If the largest possible interval in which f (x) is a decreasing function in (-3,-1) then p is equal to :

Options

A. 2
B. 6
C. 8
D. 9

Answer

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

### Step 1: Differentiate the function
First, we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^3 + 6x^2 + px + 2) = 3x^2 + 12x + p \]

### Step 2: Set the derivative less than zero
For the function to be decreasing in the interval \( (-3, -1) \), we need:
\[ f'(x) < 0 \quad \text{for } x \in (-3, -1) \]
This means we need to solve:
\[ 3x^2 + 12x + p < 0 \]

### Step 3: Find the roots of the quadratic
To determine when the quadratic is negative, we first find its roots. The roots of the quadratic \( 3x^2 + 12x + p = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 3 \), \( b = 12 \), and \( c = p \):
\[ x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 3 \cdot p}}{2 \cdot 3} = \frac{-12 \pm \sqrt{144 - 12p}}{6} \]

### Step 4: Condition for roots
For the quadratic to be negative between the roots, the roots must lie within the interval \( (-3, -1) \). Therefore, we need to ensure:
1. The left root is greater than -3.
2. The right root is less than -1.

### Step 5: Find the left root
Setting the left root greater than -3:
\[ \frac{-12 - \sqrt{144 - 12p}}{6} > -3 \]
Multiplying through by 6:
\[ -12 - \sqrt{144 - 12p} > -18 \]
Rearranging gives:
\[ \sqrt{144 - 12p} < 6 \]
Squaring both sides:
\[ 144 - 12p < 36 \]
\[ 108 < 12p \]
\[ p > 9 \]

### Step 6: Find the right root
Now, setting the right root less than -1:
\[ \frac{-12 + \sqrt{144 - 12p}}{6} < -1 \]
Multiplying through by 6:
\[ -12 + \sqrt{144 - 12p} < -6 \]
Rearranging gives:
\[ \sqrt{144 - 12p} < 6 \]
Squaring both sides:
\[ 144 - 12p < 36 \]
\[ 108 < 12p \]
\[ p > 9 \]

### Conclusion
Both conditions yield the same result. Therefore, the largest possible value of \( p \) that satisfies the conditions for \( f(x) \) to be decreasing in the interval \( (-3, -1) \) is:
\[ \boxed{9} \]