If `f(x)=2x^3+9x^2+lambdax+20` is a decreasing function fo x in the largest possible interval (-2,-1) then `lambda` =

Options

A. 12
B. -12
C. 6
D. none of these

Answer

To find the value of \(\lambda\) such that the function \(f(x) = 2x^3 + 9x^2 + \lambda x + 20\) is decreasing in the interval \((-2, -1)\), we will follow these steps:

### Step 1: Differentiate the function
To determine whether the function is increasing or decreasing, we need to find its derivative \(f'(x)\).

\[ f'(x) = \frac{d}{dx}(2x^3 + 9x^2 + \lambda x + 20) \]

Calculating the derivative:

\[ f'(x) = 6x^2 + 18x + \lambda \]

### Step 2: Set the derivative to be less than zero
For \(f(x)\) to be a decreasing function in the interval \((-2, -1)\), we need:

\[ f'(x) < 0 \quad \text{for } x \in (-2, -1) \]

This means:

\[ 6x^2 + 18x + \lambda < 0 \quad \text{for } x \in (-2, -1) \]

### Step 3: Analyze the quadratic expression
The expression \(6x^2 + 18x + \lambda\) is a quadratic function. For it to be negative in the interval \((-2, -1)\), it should have roots at \(x = -2\) and \(x = -1\). Thus, we can express it as:

\[ 6(x + 2)(x + 1) \]

### Step 4: Find the product of the roots
The product of the roots of the quadratic \(ax^2 + bx + c = 0\) is given by:

\[ \text{Product of roots} = \frac{c}{a} \]

In our case, \(c = \lambda\) and \(a = 6\). The product of the roots \(-2\) and \(-1\) is:

\[ (-2)(-1) = 2 \]

Thus, we have:

\[ \frac{\lambda}{6} = 2 \]

### Step 5: Solve for \(\lambda\)
Now, we can solve for \(\lambda\):

\[ \lambda = 2 \times 6 = 12 \]

### Conclusion
The value of \(\lambda\) is:

\[ \lambda = 12 \]

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