If 2 lies between the roots of the equation `t ^(2) - mt +2 =0,(m in R)` then the value of `[((2 |x|)/(9+x ^(2)))^(m)]` is:
(where [.] denotes greatest integer function)

Options

A. 0
B. 1
C. 8
D. 27

Answer

To solve the problem step by step, we need to analyze the given quadratic equation and the conditions provided.

### Step 1: Analyze the quadratic equation
The given quadratic equation is:
\[ t^2 - mt + 2 = 0 \]
We need to find the conditions under which 2 lies between the roots of this equation.

### Step 2: Use the properties of roots
Let the roots of the equation be \( r_1 \) and \( r_2 \). For 2 to lie between the roots, we must have:
\[ r_1 < 2 < r_2 \]
Using Vieta's formulas, we know:
- The sum of the roots \( r_1 + r_2 = m \)
- The product of the roots \( r_1 \cdot r_2 = 2 \)

### Step 3: Set up inequalities
From the conditions \( r_1 < 2 \) and \( r_2 > 2 \), we can express these in terms of \( m \):
1. \( r_1 + r_2 = m \)
2. \( r_1 \cdot r_2 = 2 \)

We can express \( r_1 \) and \( r_2 \) in terms of \( m \):
- Let \( r_1 = 2 - k \) and \( r_2 = 2 + k \) for some \( k > 0 \).

Now substituting these into the product of roots:
\[ (2 - k)(2 + k) = 2 \]
Expanding this gives:
\[ 4 - k^2 = 2 \implies k^2 = 2 \implies k = \sqrt{2} \]
Thus the roots are:
\[ r_1 = 2 - \sqrt{2}, \quad r_2 = 2 + \sqrt{2} \]
Now, using Vieta's formulas:
\[ m = r_1 + r_2 = (2 - \sqrt{2}) + (2 + \sqrt{2}) = 4 \]

### Step 4: Determine the range for \( m \)
To ensure that 2 lies between the roots, we need:
- \( m > 4 \)

### Step 5: Evaluate the expression
Now we need to evaluate the expression:
\[ \left( \frac{2 |x|}{9 + x^2} \right)^m \]
Since \( m > 4 \), we can analyze the expression for different values of \( x \).

### Step 6: Find the maximum value of the base
To find the maximum of \( \frac{2 |x|}{9 + x^2} \), we can set:
\[ y = \frac{2 |x|}{9 + x^2} \]
Taking the derivative and setting it to zero will help us find the critical points.

### Step 7: Critical points
Let \( x \geq 0 \) (since we are dealing with \( |x| \)):
\[ y = \frac{2x}{9 + x^2} \]
Using the quotient rule:
\[ y' = \frac{(9 + x^2)(2) - 2x(2x)}{(9 + x^2)^2} = \frac{18 - x^2}{(9 + x^2)^2} \]
Setting \( y' = 0 \):
\[ 18 - x^2 = 0 \implies x^2 = 18 \implies x = 3\sqrt{2} \]

### Step 8: Evaluate \( y \) at critical points
Substituting \( x = 3\sqrt{2} \):
\[ y = \frac{2(3\sqrt{2})}{9 + (3\sqrt{2})^2} = \frac{6\sqrt{2}}{9 + 18} = \frac{6\sqrt{2}}{27} = \frac{2\sqrt{2}}{9} \]

### Step 9: Evaluate the expression for \( m \)
Since \( m > 4 \), we have:
\[ \left( \frac{2\sqrt{2}}{9} \right)^m \]
As \( m \) increases, this expression approaches 0.

### Step 10: Greatest integer function
Thus, the greatest integer function of a value approaching 0 will be:
\[ \lfloor 0 \rfloor = 0 \]

### Final Answer
The value of the expression is:
\[ \boxed{0} \]