How many roots foes the following equation possess `3 ^(|x|) (|2-|x||)=2` ?

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the equation \( 3^{|x|} (|2 - |x||) = 2 \), we will follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the equation:
\[ |2 - |x|| = \frac{2}{3^{|x|}} \]

### Step 2: Analyze the expression \( |2 - |x|| \)
The expression \( |2 - |x|| \) can be broken down into two cases based on the value of \( |x| \):

1. **Case 1:** \( |x| \leq 2 \)
- In this case, \( |2 - |x|| = 2 - |x| \).
- The equation becomes:
\[ 2 - |x| = \frac{2}{3^{|x|}} \]

2. **Case 2:** \( |x| > 2 \)
- Here, \( |2 - |x|| = |x| - 2 \).
- The equation becomes:
\[ |x| - 2 = \frac{2}{3^{|x|}} \]

### Step 3: Solve Case 1
For Case 1:
\[ 2 - |x| = \frac{2}{3^{|x|}} \]
Rearranging gives:
\[ |x| = 2 - \frac{2}{3^{|x|}} \]
Let \( y = |x| \). The equation becomes:
\[ y = 2 - \frac{2}{3^y} \]

### Step 4: Analyze the function \( f(y) = y + \frac{2}{3^y} - 2 \)
We need to find the roots of \( f(y) = 0 \).

1. **Behavior of \( f(y) \)**:
- As \( y \to 0 \), \( f(0) = 0 + 2 - 2 = 0 \).
- As \( y \to \infty \), \( f(y) \to \infty \) since \( y \) dominates.
- \( f(y) \) is continuous and differentiable.

2. **Finding critical points**:
- The derivative \( f'(y) = 1 - \frac{2 \ln(3)}{3^y} \).
- Set \( f'(y) = 0 \) to find critical points:
\[ 1 = \frac{2 \ln(3)}{3^y} \implies 3^y = 2 \ln(3) \implies y = \log_3(2 \ln(3)) \]

3. **Number of roots**:
- Check the behavior of \( f(y) \) around the critical point to determine the number of roots.

### Step 5: Solve Case 2
For Case 2:
\[ |x| - 2 = \frac{2}{3^{|x|}} \]
Rearranging gives:
\[ |x| = 2 + \frac{2}{3^{|x|}} \]
Let \( z = |x| \). The equation becomes:
\[ z = 2 + \frac{2}{3^z} \]

### Step 6: Analyze the function \( g(z) = z - 2 - \frac{2}{3^z} \)
1. **Behavior of \( g(z) \)**:
- As \( z \to 2 \), \( g(2) = 0 \).
- As \( z \to \infty \), \( g(z) \to \infty \).

2. **Finding critical points**:
- The derivative \( g'(z) = 1 + \frac{2 \ln(3)}{3^z} \) is always positive, indicating \( g(z) \) is increasing.

3. **Number of roots**:
- Since \( g(z) \) is increasing and crosses the x-axis at \( z = 2 \), there is exactly one root in this case.

### Conclusion
Combining the results from both cases, we find that:

- Case 1 can yield up to 2 roots.
- Case 2 yields 1 root.

Thus, the total number of roots for the equation \( 3^{|x|} (|2 - |x||) = 2 \) is **3**.