Let x, y, z be elements from interval `[0,2pi] ` satisfying the inequality ` (4+ sin 4 x )(2+ cot^(2) y)(1+ sin^(4) z) le 12 sin^(2) z ` , then

Options

A. the number of ordered pairs (x,y) is 5
B. the number of ordered pairs (y,z) is 8
C. the number of ordered pairs (z,x) is 8
D. the number of pairs (y,z) such that z=y is 2

Answer

To solve the inequality \( (4 + \sin 4x)(2 + \cot^2 y)(1 + \sin^4 z) \leq 12 \sin^2 z \), we will analyze each component of the inequality step by step.

### Step 1: Analyze the components of the inequality

1. **Understanding the range of \( \sin 4x \)**:
- The function \( \sin 4x \) oscillates between -1 and 1.
- Therefore, \( 4 + \sin 4x \) will oscillate between \( 4 - 1 = 3 \) and \( 4 + 1 = 5 \).
- Thus, we have:
\[ 3 \leq 4 + \sin 4x \leq 5 \]

2. **Understanding the range of \( \cot^2 y \)**:
- The function \( \cot^2 y \) is non-negative and approaches infinity as \( y \) approaches \( 0 \) or \( \pi \).
- Therefore, \( 2 + \cot^2 y \) is always greater than or equal to 2:
\[ 2 \leq 2 + \cot^2 y \]

3. **Understanding the range of \( \sin^4 z \)**:
- The function \( \sin^4 z \) oscillates between 0 and 1.
- Therefore, \( 1 + \sin^4 z \) will oscillate between 1 and 2:
\[ 1 \leq 1 + \sin^4 z \leq 2 \]

### Step 2: Combine the inequalities

Now, we can combine these inequalities into the original inequality:

\[ (4 + \sin 4x)(2 + \cot^2 y)(1 + \sin^4 z) \leq 12 \sin^2 z \]

Using the lower bounds, we have:

\[ 3 \cdot 2 \cdot 1 \leq 12 \sin^2 z \]

This simplifies to:

\[ 6 \leq 12 \sin^2 z \]

Dividing both sides by 12 gives:

\[ \frac{1}{2} \leq \sin^2 z \]

Thus, we find:

\[ \sin z \geq \frac{\sqrt{2}}{2} \quad \text{or} \quad \sin z \leq -\frac{\sqrt{2}}{2} \]

### Step 3: Determine the values of \( z \)

The solutions for \( z \) in the interval \([0, 2\pi]\) are:

- For \( \sin z \geq \frac{\sqrt{2}}{2} \):
- \( z = \frac{\pi}{4}, \frac{3\pi}{4} \) (in the first and second quadrants)

- For \( \sin z \leq -\frac{\sqrt{2}}{2} \):
- \( z = \frac{5\pi}{4}, \frac{7\pi}{4} \) (in the third and fourth quadrants)

Thus, the possible values of \( z \) are:

\[ z = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \]

### Step 4: Determine the values of \( y \)

From the inequality \( \cot^2 y = 0 \), we find:

\[ y = \frac{\pi}{2}, \frac{3\pi}{2} \]

### Step 5: Determine the values of \( x \)

From \( \sin 4x = -1 \):

\[ 4x = \frac{3\pi}{2} + 2k\pi \quad \text{for } k \in \mathbb{Z} \]

This gives:

\[ x = \frac{3\pi}{8} + \frac{k\pi}{2} \quad \text{for } k = 0, 1, 2, 3 \]

The valid values of \( x \) in the interval \([0, 2\pi]\) are:

- \( x = \frac{3\pi}{8}, \frac{7\pi}{8}, \frac{11\pi}{8}, \frac{15\pi}{8} \)

### Step 6: Count the ordered pairs

1. **Ordered pairs \( (x, y) \)**:
- \( x \) has 4 values and \( y \) has 2 values:
- Total pairs: \( 4 \times 2 = 8 \)

2. **Ordered pairs \( (y, z) \)**:
- \( y \) has 2 values and \( z \) has 4 values:
- Total pairs: \( 2 \times 4 = 8 \)

3. **Ordered pairs \( (x, z) \)**:
- \( x \) has 4 values and \( z \) has 4 values:
- Total pairs: \( 4 \times 4 = 16 \)

4. **Pairs \( (y, z) \) such that \( z = y \)**:
- The only valid pairs are \( (y, z) = \left(\frac{\pi}{2}, \frac{\pi}{2}\right) \) and \( \left(\frac{3\pi}{2}, \frac{3\pi}{2}\right) \).
- Total pairs: 2

### Final Results

- Number of ordered pairs \( (x, y) \): 8
- Number of ordered pairs \( (y, z) \): 8
- Number of ordered pairs \( (x, z) \): 16
- Number of pairs \( (y, z) \) such that \( z = y \): 2