If 1 lies between the roots of equation `y^2 - my +1 = 0` and [x] denotes the integral part of x, then `[((4|x|)/(x^2+16))^m]`where `x in R` is equal to

Options

A. 0
B. 1
C. 2
D. undefined

Explanation

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

### Step 1: Analyze the quadratic equation
The given quadratic equation is:
\[ y^2 - my + 1 = 0 \]

### Step 2: Condition for roots
We know that 1 lies between the roots of the quadratic equation. For this to happen, two conditions must be satisfied:
1. The discriminant must be greater than 0 (indicating two distinct roots).
2. The value of the quadratic at \( y = 1 \) must be less than 0 (indicating that 1 is between the roots).

### Step 3: Calculate the discriminant
The discriminant \( D \) of the quadratic equation is given by:
\[ D = m^2 - 4 \cdot 1 \cdot 1 = m^2 - 4 \]
For the roots to be real and distinct, we require:
\[ D > 0 \]
Thus, we have:
\[ m^2 - 4 > 0 \]
This implies:
\[ m > 2 \quad \text{or} \quad m < -2 \]

### Step 4: Evaluate the function at \( y = 1 \)
Next, we evaluate the quadratic at \( y = 1 \):
\[ f(1) = 1^2 - m \cdot 1 + 1 = 2 - m \]
For 1 to lie between the roots, we need:
\[ f(1) < 0 \]
Thus:
\[ 2 - m < 0 \]
This implies:
\[ m > 2 \]

### Step 5: Combine the conditions
From the two conditions derived:
1. \( m > 2 \)
2. \( m < -2 \)

The common area is:
\[ m > 2 \]

### Step 6: Analyze the expression
We need to evaluate:
\[ \left[ \left( \frac{4 |x|}{x^2 + 16} \right)^m \right] \]
where \([x]\) denotes the integral part of \(x\).

### Step 7: Find the range of the expression
Since \( x^2 + 16 \) is always positive, we can analyze the expression:
\[ \frac{4 |x|}{x^2 + 16} \]
- The maximum value occurs when \( x = 0 \):
\[ \frac{4 \cdot 0}{0^2 + 16} = 0 \]
- As \( |x| \) increases, the value of \( \frac{4 |x|}{x^2 + 16} \) approaches 0.

### Step 8: Determine the limits
The expression \( \frac{4 |x|}{x^2 + 16} \) is always positive and less than or equal to 1 for all \( x \) in \( \mathbb{R} \). Therefore, when raised to the power \( m \) (where \( m > 2 \)), the expression will also be between 0 and 1.

### Step 9: Evaluate the integral part
Since:
\[ 0 < \left( \frac{4 |x|}{x^2 + 16} \right)^m < 1 \]
The integral part of any number between 0 and 1 is:
\[ \left[ \left( \frac{4 |x|}{x^2 + 16} \right)^m \right] = 0 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{0} \]