If `alpha, beta` be the roots of `4x^(8) - 16x + c = 0, c in R` such that `1 lt alpha lt 2 and 2 lt beta lt 3`, then the number of integral values of c is

Explanation

To solve the problem, we need to analyze the quadratic equation given by:

\[ 4x^2 - 16x + c = 0 \]

where \( \alpha \) and \( \beta \) are the roots such that \( 1 < \alpha < 2 \) and \( 2 < \beta < 3 \). We will determine the conditions on \( c \) based on the values of the function at specific points.

### Step 1: Evaluate the function at \( x = 1 \)

We start by substituting \( x = 1 \) into the equation:

\[ f(1) = 4(1)^2 - 16(1) + c = 4 - 16 + c = c - 12 \]

Since \( \alpha \) is a root and lies between 1 and 2, we require:

\[ f(1) > 0 \implies c - 12 > 0 \implies c > 12 \]

**Hint:** Evaluate the function at the lower boundary of the root interval.

### Step 2: Evaluate the function at \( x = 2 \)

Next, we substitute \( x = 2 \):

\[ f(2) = 4(2)^2 - 16(2) + c = 16 - 32 + c = c - 16 \]

Since \( \alpha < 2 \) and \( \beta > 2 \), we require:

\[ f(2) < 0 \implies c - 16 < 0 \implies c < 16 \]

**Hint:** Evaluate the function at the upper boundary of the root interval.

### Step 3: Evaluate the function at \( x = 3 \)

Now, we substitute \( x = 3 \):

\[ f(3) = 4(3)^2 - 16(3) + c = 36 - 48 + c = c - 12 \]

Since \( \beta < 3 \), we require:

\[ f(3) > 0 \implies c - 12 > 0 \implies c > 12 \]

**Hint:** Ensure that the function is positive at the upper boundary of the root interval.

### Step 4: Combine the inequalities

From the evaluations, we have established the following inequalities:

1. \( c > 12 \)
2. \( c < 16 \)

Combining these gives:

\[ 12 < c < 16 \]

### Step 5: Determine integral values of \( c \)

The integral values of \( c \) that satisfy \( 12 < c < 16 \) are:

- \( c = 13 \)
- \( c = 14 \)
- \( c = 15 \)

Thus, the number of integral values of \( c \) is:

\[ \text{Number of integral values of } c = 3 \]

**Final Answer:** The number of integral values of \( c \) is 3.

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