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, we need to determine the value of the expression \(\left[\left(\frac{2 |x|}{9 + x^2}\right)^{m}\right]\), given that \(2\) lies between the roots of the quadratic equation \(t^2 - mt + 2 = 0\).

### Step 1: Identify the roots of the quadratic equation
The roots of the quadratic equation \(t^2 - mt + 2 = 0\) can be found using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -m\), and \(c = 2\). Thus, the roots are:
\[ t = \frac{m \pm \sqrt{m^2 - 8}}{2} \]

### Step 2: Condition for \(2\) to lie between the roots
For \(2\) to lie between the roots, we need:
\[ \text{min(root1, root2)} < 2 < \text{max(root1, root2)} \]
This implies:
\[ \frac{m - \sqrt{m^2 - 8}}{2} < 2 < \frac{m + \sqrt{m^2 - 8}}{2} \]

### Step 3: Solve the inequalities
1. From the left inequality:
\[ \frac{m - \sqrt{m^2 - 8}}{2} < 2 \]
Multiplying through by 2:
\[ m - \sqrt{m^2 - 8} < 4 \]
Rearranging gives:
\[ \sqrt{m^2 - 8} > m - 4 \]
Squaring both sides:
\[ m^2 - 8 > (m - 4)^2 \]
Expanding the right side:
\[ m^2 - 8 > m^2 - 8m + 16 \]
Simplifying:
\[ 0 > -8m + 24 \quad \Rightarrow \quad 8m > 24 \quad \Rightarrow \quad m > 3 \]

2. From the right inequality:
\[ 2 < \frac{m + \sqrt{m^2 - 8}}{2} \]
Multiplying through by 2:
\[ 4 < m + \sqrt{m^2 - 8} \]
Rearranging gives:
\[ \sqrt{m^2 - 8} > 4 - m \]
Squaring both sides:
\[ m^2 - 8 > (4 - m)^2 \]
Expanding the right side:
\[ m^2 - 8 > 16 - 8m + m^2 \]
Simplifying:
\[ -8 > 16 - 8m \quad \Rightarrow \quad -24 > -8m \quad \Rightarrow \quad m > 3 \]

### Step 4: Conclusion on the value of \(m\)
From both inequalities, we conclude:
\[ m > 3 \]

### Step 5: Evaluate the expression
Now we need to evaluate \(\left[\left(\frac{2 |x|}{9 + x^2}\right)^{m}\right]\).

1. The expression \(\frac{2 |x|}{9 + x^2}\) is maximized when \(x\) is chosen appropriately.
2. To find the maximum value, we can differentiate \(\frac{2x}{9+x^2}\) and set the derivative to zero. However, we can also analyze the behavior of the function:
- As \(x \to 0\), \(\frac{2 |x|}{9 + x^2} \to 0\).
- As \(x \to \infty\), \(\frac{2 |x|}{9 + x^2} \to 0\).
- The maximum occurs at a finite value of \(x\).

### Step 6: Find the maximum value
By testing values, we find that:
\[ \frac{2 |x|}{9 + x^2} \leq \frac{2 \cdot 3}{9 + 3^2} = \frac{6}{18} = \frac{1}{3} \]
Thus, the maximum value of \(\frac{2 |x|}{9 + x^2}\) is \(\frac{1}{3}\).

### Step 7: Substitute into the expression
Now substituting back:
\[ \left(\frac{2 |x|}{9 + x^2}\right)^{m} \leq \left(\frac{1}{3}\right)^{m} \]
Since \(m > 3\), we have:
\[ \left(\frac{1}{3}\right)^{m} < \left(\frac{1}{3}\right)^{3} = \frac{1}{27} \]
This implies that:
\[ \left[\left(\frac{2 |x|}{9 + x^2}\right)^{m}\right] = 0 \]

### Final Answer
Thus, the value of \(\left[\left(\frac{2 |x|}{9 + x^2}\right)^{m}\right]\) is:
\[ \boxed{0} \]