If `f(x)=x^3+4x^2+lambdax+1` is a monotonically decreasing function of `x` in the largest possible interval `(-2,-2/3)dot` Then (a)`lambda=4` (b) `lambda=2` `lambda=-1` (d) `lambda` has no real value

Explanation

To determine the value of \(\lambda\) such that the function \(f(x) = x^3 + 4x^2 + \lambda x + 1\) is monotonically decreasing in the interval \((-2, -\frac{2}{3})\), we need to analyze the derivative of the function.

### Step 1: Find the derivative of the function
The first step is to find the derivative \(f'(x)\):

\[ f'(x) = \frac{d}{dx}(x^3) + \frac{d}{dx}(4x^2) + \frac{d}{dx}(\lambda x) + \frac{d}{dx}(1) \]
...