Let `f:Rrarr R` be defined as `f(x)=sin(|x|)`
Which one of the following is correct?

Options

A. A) f is not differentiable only at 0
B. B) f is differentiable at 0 only
C. C) f is differentiable everywhere
D. D) f is non - differentiable at many points

Answer

To solve the problem, we need to analyze the function \( f(x) = \sin(|x|) \) and determine its differentiability, particularly at the critical point \( x = 0 \).

### Step-by-Step Solution:

1. **Define the Function**:
The function is given as \( f(x) = \sin(|x|) \). We need to consider the behavior of this function for different values of \( x \).

2. **Break the Function into Cases**:
- For \( x \geq 0 \):
\[ f(x) = \sin(x) \]
- For \( x < 0 \):
\[ f(x) = \sin(-x) = -\sin(x) \]

3. **Check Differentiability at \( x = 0 \)**:
To determine if the function is differentiable at \( x = 0 \), we need to calculate the left-hand derivative and the right-hand derivative at this point.

4. **Calculate the Left-Hand Derivative**:
The left-hand derivative at \( x = 0 \) is given by:
\[ f'(0^-) = \lim_{h \to 0^-} \frac{f(0 + h) - f(0)}{h} \]
Substituting \( f(0) = \sin(0) = 0 \):
\[ f'(0^-) = \lim_{h \to 0^-} \frac{f(h) - 0}{h} = \lim_{h \to 0^-} \frac{-\sin(-h)}{h} = \lim_{h \to 0^-} \frac{\sin(h)}{-h} \]
This limit evaluates to:
\[ f'(0^-) = -1 \]

5. **Calculate the Right-Hand Derivative**:
The right-hand derivative at \( x = 0 \) is given by:
\[ f'(0^+) = \lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} \]
Again substituting \( f(0) = 0 \):
\[ f'(0^+) = \lim_{h \to 0^+} \frac{f(h) - 0}{h} = \lim_{h \to 0^+} \frac{\sin(h)}{h} \]
This limit evaluates to:
\[ f'(0^+) = 1 \]

6. **Compare the Left-Hand and Right-Hand Derivatives**:
We have:
\[ f'(0^-) = -1 \quad \text{and} \quad f'(0^+) = 1 \]
Since the left-hand derivative does not equal the right-hand derivative, we conclude that \( f(x) \) is not differentiable at \( x = 0 \).

7. **Conclusion**:
The function \( f(x) = \sin(|x|) \) is differentiable everywhere except at \( x = 0 \).

### Final Answer:
The correct statement is that \( f \) is not differentiable only at the point \( 0 \).