The number of real solutions of the equation `e^(|x|)-|x|=0`, is
A. o
B. 1
c. 2
D. None of these.

Options

A. 0
B. 1
C. 2
D. None of these

Answer

To find the number of real solutions for the equation \( e^{|x|} - |x| = 0 \), we can break down the problem into steps.

### Step 1: Rewrite the equation
We start with the equation:
\[ e^{|x|} = |x| \]

### Step 2: Analyze cases based on the absolute value
We need to consider two cases based on the definition of absolute value.

**Case 1:** \( x \geq 0 \)
In this case, \( |x| = x \). The equation simplifies to:
\[ e^x = x \]

**Case 2:** \( x < 0 \)
Here, \( |x| = -x \). The equation becomes:
\[ e^{-x} = -x \]

### Step 3: Analyze Case 1
For the equation \( e^x = x \):
- The function \( e^x \) is an exponential function that is always positive and increases rapidly.
- The function \( x \) is a linear function that passes through the origin.

To find the intersection points, we can analyze the behavior of these functions:
- At \( x = 0 \), \( e^0 = 1 \) and \( 0 = 0 \). So, \( e^x > x \) for \( x = 0 \).
- As \( x \) increases, \( e^x \) grows faster than \( x \).
- Therefore, there are no points where \( e^x = x \) for \( x \geq 0 \).

### Step 4: Analyze Case 2
For the equation \( e^{-x} = -x \):
- The function \( e^{-x} \) is positive for all \( x \), while \( -x \) is negative for \( x < 0 \).
- As \( x \) approaches negative infinity, \( e^{-x} \) approaches infinity and \( -x \) approaches positive infinity.

To find the intersection points:
- At \( x = -1 \), \( e^{-(-1)} = e^1 = e \) and \( -(-1) = 1 \). Since \( e > 1 \), we have \( e^{-x} > -x \).
- As \( x \) decreases further (more negative), \( e^{-x} \) will continue to grow larger than \( -x \).

### Step 5: Conclusion
From the analysis of both cases, we find that:
- There are no solutions for \( x \geq 0 \).
- There are no solutions for \( x < 0 \) since \( e^{-x} \) will always be greater than \( -x \).

Thus, the number of real solutions to the equation \( e^{|x|} - |x| = 0 \) is:
\[ \text{Number of real solutions} = 0 \]

### Final Answer
The answer is **A. 0**.