The number of solutions of the equation `2^|x|=1+2|cosx|` is

Options

A. (a)0
B. (b)2
C. (c)4
D. (d)Infinite

Explanation

To solve the equation \( 2^{|x|} = 1 + 2|\cos x| \), we will analyze both sides of the equation step by step.

### Step 1: Analyze the left-hand side
The left-hand side of the equation is \( 2^{|x|} \). This function is always positive and increases as \( |x| \) increases.

**Hint:** Consider the behavior of exponential functions and how they grow.

### Step 2: Analyze the right-hand side
The right-hand side of the equation is \( 1 + 2|\cos x| \). The function \( |\cos x| \) oscillates between 0 and 1, hence \( 1 + 2|\cos x| \) oscillates between 1 and 3.

**Hint:** Remember that the maximum value of \( |\cos x| \) is 1, which will help you find the range of the right-hand side.

### Step 3: Determine the ranges
From the analysis:
- The left-hand side \( 2^{|x|} \) can take values from \( 1 \) to \( \infty \) as \( |x| \) increases.
- The right-hand side \( 1 + 2|\cos x| \) takes values from \( 1 \) to \( 3 \).

**Hint:** Identify the points where both sides can be equal based on their ranges.

### Step 4: Find intersections
To find the number of solutions, we need to find the points where \( 2^{|x|} \) intersects \( 1 + 2|\cos x| \).

1. **At \( |x| = 0 \):**
\[ 2^{|0|} = 2^0 = 1 \]
\[ 1 + 2|\cos(0)| = 1 + 2 \cdot 1 = 3 \]
So, \( 1 = 3 \) does not hold.

2. **At \( |x| = 1 \):**
\[ 2^{|1|} = 2^1 = 2 \]
\[ 1 + 2|\cos(1)| \text{ (where } \cos(1) \text{ is approximately } 0.5403) \]
\[ 1 + 2 \cdot 0.5403 \approx 2.0806 \]
So, \( 2 \approx 2.0806 \) does not hold.

3. **At \( |x| = \pi/2 \):**
\[ 2^{|\pi/2|} \text{ (which is a value greater than 2)} \]
\[ 1 + 2|\cos(\pi/2)| = 1 + 2 \cdot 0 = 1 \]
So, \( 2^{|\pi/2|} > 1 \).

4. **At \( |x| = \pi \):**
\[ 2^{|\pi|} \text{ (which is a value greater than 3)} \]
\[ 1 + 2|\cos(\pi)| = 1 + 2 \cdot (-1) = -1 \]
So, \( 2^{|\pi|} > -1 \).

### Step 5: Conclusion on intersections
The function \( 2^{|x|} \) will intersect the oscillating function \( 1 + 2|\cos x| \) at two points within the range of \( 1 \) to \( 3 \) as \( |x| \) increases.

Thus, the total number of solutions to the equation \( 2^{|x|} = 1 + 2|\cos x| \) is **2**.

### Final Answer
The number of solutions is **2**.

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