Let ` alpha = max_(x sub R) { 8^(2 sin 3 x) * 4^( 4 cos 3 x)} and beta = min_(x sub R) { 8 ^(2 sin 3 x) * 4 ^(4 cos 3x)} ` . If ` 8 x^(2) + bx + c = 0 ` is a quadratic equation whose roots are ` alpha ^(1//5) and beta ^(1//5)` then the value of `c – b` is equal to :

Options

A. 42
B. 47
C. 43
D. 50

Answer

To solve the problem, we need to find the values of \( \alpha \) and \( \beta \) based on the given expressions and then use these values to find \( c - b \) from the quadratic equation.

### Step 1: Express \( \alpha \) and \( \beta \)

Given:
\[ \alpha = \max_{x \in \mathbb{R}} \{ 8^{2 \sin 3x} \cdot 4^{4 \cos 3x} \} \]
\[ \beta = \min_{x \in \mathbb{R}} \{ 8^{2 \sin 3x} \cdot 4^{4 \cos 3x} \} \]

We can rewrite \( 8 \) and \( 4 \) in terms of base \( 2 \):
\[ 8 = 2^3 \quad \text{and} \quad 4 = 2^2 \]
Thus,
\[ 8^{2 \sin 3x} = (2^3)^{2 \sin 3x} = 2^{6 \sin 3x} \]
\[ 4^{4 \cos 3x} = (2^2)^{4 \cos 3x} = 2^{8 \cos 3x} \]
Combining these, we have:
\[ 8^{2 \sin 3x} \cdot 4^{4 \cos 3x} = 2^{6 \sin 3x + 8 \cos 3x} \]

### Step 2: Find the maximum and minimum values

Now we need to find the maximum and minimum of the expression \( 6 \sin 3x + 8 \cos 3x \). This can be expressed in the form \( R \sin(3x + \phi) \), where:
\[ R = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]
The maximum value of \( 6 \sin 3x + 8 \cos 3x \) is \( 10 \) and the minimum value is \( -10 \).

Thus,
\[ \alpha = 2^{10} = 1024 \]
\[ \beta = 2^{-10} = \frac{1}{1024} \]

### Step 3: Find the roots of the quadratic equation

The roots of the quadratic equation \( 8x^2 + bx + c = 0 \) are given as \( \alpha^{1/5} \) and \( \beta^{1/5} \):
\[ \alpha^{1/5} = (1024)^{1/5} = 4 \]
\[ \beta^{1/5} = \left(\frac{1}{1024}\right)^{1/5} = \frac{1}{4} \]

### Step 4: Use Vieta's formulas

From Vieta's formulas, we know:
- The sum of the roots \( \alpha^{1/5} + \beta^{1/5} = 4 + \frac{1}{4} = \frac{16 + 1}{4} = \frac{17}{4} \)
- The product of the roots \( \alpha^{1/5} \cdot \beta^{1/5} = 4 \cdot \frac{1}{4} = 1 \)

Using Vieta's relations:
1. The sum of the roots gives us:
\[ -\frac{b}{8} = \frac{17}{4} \implies b = -8 \cdot \frac{17}{4} = -34 \]
2. The product of the roots gives us:
\[ \frac{c}{8} = 1 \implies c = 8 \]

### Step 5: Calculate \( c - b \)

Finally, we find:
\[ c - b = 8 - (-34) = 8 + 34 = 42 \]

### Final Answer
Thus, the value of \( c - b \) is \( \boxed{42} \).