The function f(x) `=e^(|x|)` is

Options

A. continuous everywhere but not differentiable at x=0
B. continuous and differentiable everywhere
C. not continuous at x=0
D. none of these

Answer

To determine the properties of the function \( f(x) = e^{|x|} \), we will analyze its continuity and differentiability.

### Step 1: Understanding the Function
The function \( f(x) = e^{|x|} \) involves the absolute value function, which can be expressed piecewise:
- For \( x \geq 0 \), \( |x| = x \), so \( f(x) = e^x \).
- For \( x < 0 \), \( |x| = -x \), so \( f(x) = e^{-x} \).

### Step 2: Check for Continuity
To check if \( f(x) \) is continuous everywhere, we need to verify its continuity at \( x = 0 \) and for \( x \neq 0 \).

1. **For \( x \neq 0 \)**:
- Both \( e^x \) and \( e^{-x} \) are continuous functions. Therefore, \( f(x) \) is continuous for all \( x \neq 0 \).

2. **At \( x = 0 \)**:
- We need to check the limit as \( x \) approaches 0 from both sides:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} e^{-x} = e^0 = 1 \]
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} e^x = e^0 = 1 \]
- Since \( f(0) = e^{|0|} = e^0 = 1 \), we have:
\[ \lim_{x \to 0} f(x) = f(0) \]
- Thus, \( f(x) \) is continuous at \( x = 0 \).

### Conclusion on Continuity
Since \( f(x) \) is continuous for all \( x \), we conclude that \( f(x) \) is continuous everywhere.

### Step 3: Check for Differentiability
Next, we check if \( f(x) \) is differentiable at \( x = 0 \).

1. **For \( x > 0 \)**:
- The derivative is:
\[ f'(x) = \frac{d}{dx}(e^x) = e^x \]

2. **For \( x < 0 \)**:
- The derivative is:
\[ f'(x) = \frac{d}{dx}(e^{-x}) = -e^{-x} \]

3. **At \( x = 0 \)**:
- We need to check the left-hand and right-hand derivatives:
\[ f'(0^+) = \lim_{h \to 0^+} \frac{f(h) - f(0)}{h} = \lim_{h \to 0^+} \frac{e^h - 1}{h} = e^0 = 1 \]
\[ f'(0^-) = \lim_{h \to 0^-} \frac{f(h) - f(0)}{h} = \lim_{h \to 0^-} \frac{e^{-h} - 1}{h} = -1 \]
- Since \( f'(0^+) \neq f'(0^-) \), the derivative does not exist at \( x = 0 \).

### Conclusion on Differentiability
The function \( f(x) = e^{|x|} \) is continuous everywhere but not differentiable at \( x = 0 \).

### Final Answer
Thus, the correct option is:
- Continuous everywhere but not differentiable at \( x = 0 \).

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