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

Options

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

Answer

To solve the equation \( \cos x + |x| = 0 \), we will analyze the equation step by step.

### Step 1: Rewrite the equation
The given equation is:
\[ \cos x + |x| = 0 \]
This can be rewritten as:
\[ \cos x = -|x| \]

### Step 2: Analyze the cases for \( |x| \)
The absolute value function \( |x| \) has two cases:
1. When \( x \geq 0 \), \( |x| = x \)
2. When \( x < 0 \), \( |x| = -x \)

### Step 3: Case 1: \( x \geq 0 \)
In this case, we substitute \( |x| \) with \( x \):
\[ \cos x = -x \]
Now, we need to analyze the functions \( \cos x \) and \( -x \):
- The function \( \cos x \) oscillates between 1 and -1.
- The function \( -x \) is a straight line that decreases from 0 to negative infinity as \( x \) increases.

Since \( \cos x \) can only take values between -1 and 1, the equation \( \cos x = -x \) can only have solutions for \( x \) in the interval [0, 1].

### Step 4: Finding intersections in Case 1
We need to check if there are any intersections in the interval [0, 1]:
- At \( x = 0 \): \( \cos(0) = 1 \) and \( -0 = 0 \) (no intersection).
- At \( x = 1 \): \( \cos(1) \) is approximately 0.54 and \( -1 = -1 \) (no intersection).

Since \( \cos x \) is decreasing and \( -x \) is also decreasing, and they do not intersect in the interval [0, 1], we conclude that there are no solutions for \( x \geq 0 \).

### Step 5: Case 2: \( x < 0 \)
In this case, we substitute \( |x| \) with \( -x \):
\[ \cos x = -(-x) \implies \cos x = x \]
Now we need to analyze the functions \( \cos x \) and \( x \):
- The function \( \cos x \) oscillates between 1 and -1.
- The function \( x \) is a straight line that increases from negative infinity to 0 as \( x \) increases.

### Step 6: Finding intersections in Case 2
We need to check if there are any intersections for \( x < 0 \):
- At \( x = -1 \): \( \cos(-1) \) is approximately 0.54 and \( -1 = -1 \) (no intersection).
- At \( x = 0 \): \( \cos(0) = 1 \) and \( 0 = 0 \) (no intersection).

Since \( \cos x \) is decreasing and \( x \) is increasing, and they do not intersect in the interval \( (-\infty, 0) \), we conclude that there are no solutions for \( x < 0 \).

### Conclusion
Since there are no solutions in both cases, we conclude that the number of solutions to the equation \( \cos x + |x| = 0 \) is:
\[ \text{Number of solutions} = 0 \]