If the equation `x^(3)-6x^(2)+9x+lambda=0` has exactly one root in (1, 3), then `lambda` belongs to the interval

Options

A. `(-6, -3)`
B. `(-4, 0)`
C. `(-2, 2)`
D. `(-1, 3)`

Answer

To find the values of \(\lambda\) for which the equation \(x^3 - 6x^2 + 9x + \lambda = 0\) has exactly one root in the interval \((1, 3)\), we can follow these steps:

### Step 1: Define the function
Let \(f(x) = x^3 - 6x^2 + 9x + \lambda\).

### Step 2: Evaluate the function at the endpoints of the interval
We need to evaluate \(f(1)\) and \(f(3)\):

1. **Calculate \(f(1)\)**:
\[ f(1) = 1^3 - 6(1^2) + 9(1) + \lambda = 1 - 6 + 9 + \lambda = 4 + \lambda \]

2. **Calculate \(f(3)\)**:
\[ f(3) = 3^3 - 6(3^2) + 9(3) + \lambda = 27 - 54 + 27 + \lambda = 0 + \lambda = \lambda \]

### Step 3: Set conditions for exactly one root
For the cubic equation to have exactly one root in the interval \((1, 3)\), we need:
- \(f(1) < 0\) (which means \(4 + \lambda < 0\))
- \(f(3) > 0\) (which means \(\lambda > 0\))

### Step 4: Solve the inequalities
1. From \(4 + \lambda < 0\):
\[ \lambda < -4 \]

2. From \(\lambda > 0\):
\[ \lambda > 0 \]

### Step 5: Combine the conditions
The conditions \( \lambda < -4\) and \(\lambda > 0\) cannot be satisfied simultaneously. Therefore, we need to analyze the behavior of the function further.

### Step 6: Analyze the derivative
To find the critical points, we calculate the derivative:
\[ f'(x) = 3x^2 - 12x + 9 \]
Setting \(f'(x) = 0\) gives:
\[ 3(x^2 - 4x + 3) = 0 \implies (x-3)(x-1) = 0 \]
Thus, \(x = 1\) and \(x = 3\) are critical points.

### Step 7: Determine the nature of the critical points
- \(f(1) = 4 + \lambda\)
- \(f(3) = \lambda\)

For the function to have exactly one root in \((1, 3)\), we need:
- \(f(1) < 0\) (i.e., \(4 + \lambda < 0 \Rightarrow \lambda < -4\))
- \(f(3) > 0\) (i.e., \(\lambda > 0\))

However, since these conditions contradict, we need to check the values of \(\lambda\) that allow for a double root at either endpoint.

### Step 8: Find the interval for \(\lambda\)
To ensure that there is exactly one root in the interval \((1, 3)\), we can set:
- \(f(1) = 0\) and \(f(3) = 0\) to find the values of \(\lambda\).

1. **Setting \(f(1) = 0\)**:
\[ 4 + \lambda = 0 \implies \lambda = -4 \]

2. **Setting \(f(3) = 0\)**:
\[ \lambda = 0 \]

### Final Conclusion
Thus, for \(\lambda\) to allow exactly one root in the interval \((1, 3)\), \(\lambda\) must lie in the interval:
\[ \lambda \in (-4, 0) \]