The number of solutions of the equation
`|x|-cos x=0`, is

Options

A. 1
B. 2
C. 3
D. 0

Explanation

To find the number of solutions for the equation \(|x| - \cos x = 0\), we can rewrite it as:

\[ |x| = \cos x \]

### Step 1: Understand the functions involved
We need to analyze the two functions: \(y = |x|\) and \(y = \cos x\).

### Step 2: Graph the functions
1. **Graph of \(y = |x|\)**:
- This is a V-shaped graph that opens upwards.
- For \(x < 0\), \(y = -x\) (line with a slope of -1).
- For \(x \geq 0\), \(y = x\) (line with a slope of 1).

2. **Graph of \(y = \cos x\)**:
- This is a wave-like graph oscillating between -1 and 1.
- It has a period of \(2\pi\) and intersects the y-axis at (0,1).

### Step 3: Find the intersection points
To find the number of solutions, we need to determine where the graphs of \(y = |x|\) and \(y = \cos x\) intersect.

- For \(x \geq 0\):
- The equation becomes \(x = \cos x\).

- For \(x < 0\):
- The equation becomes \(-x = \cos x\) or \(x = -\cos x\).

### Step 4: Analyze the intersections
1. **For \(x \geq 0\)**:
- The line \(y = x\) intersects the curve \(y = \cos x\) at some points.
- Since \(\cos x\) starts at 1 when \(x = 0\) and decreases to 0 at \(x = \frac{\pi}{2}\), there will be at least one intersection in this interval.
- As \(x\) continues to increase, \(\cos x\) oscillates, and we can find additional intersections.

2. **For \(x < 0\)**:
- The line \(y = -x\) intersects the curve \(y = \cos x\) similarly.
- Since \(\cos x\) is also positive in certain intervals, there will be intersections in this region as well.

### Step 5: Count the solutions
By analyzing the graphs:
- The function \(y = \cos x\) oscillates between -1 and 1, while \(y = |x|\) increases without bound.
- The intersections will occur where the oscillating function meets the linear function.

From the graphical analysis:
- We find that there are **two intersections** for \(x \geq 0\) and **two intersections** for \(x < 0\).

### Conclusion
Thus, the total number of solutions to the equation \(|x| - \cos x = 0\) is:

\[ \text{Total Solutions} = 2 + 2 = 4 \]

### Final Answer
The number of solutions of the equation \(|x| - \cos x = 0\) is **4**.

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