Domain of differentiations of the function f(x) = |x -2| cos x is

Options

A. R
B. `R - {2}`
C. `(0 , oo)`
D. {2}

Answer

To find the domain of differentiability of the function \( f(x) = |x - 2| \cos x \), we need to analyze the components of the function and where they are defined.

### Step 1: Identify the components of the function
The function consists of two parts:
1. \( |x - 2| \)
2. \( \cos x \)

### Step 2: Determine the domain of each component
- The absolute value function \( |x - 2| \) is defined for all real numbers \( x \).
- The cosine function \( \cos x \) is also defined for all real numbers \( x \).

### Step 3: Consider the differentiability of \( |x - 2| \)
The function \( |x - 2| \) is differentiable everywhere except at the point where its argument is zero, which is at \( x = 2 \). At this point, the function has a sharp corner, and thus the derivative does not exist.

### Step 4: Combine the results
Since \( \cos x \) is differentiable everywhere, the differentiability of the entire function \( f(x) = |x - 2| \cos x \) will be affected by the non-differentiability of \( |x - 2| \) at \( x = 2 \).

### Conclusion
The domain of differentiability of the function \( f(x) = |x - 2| \cos x \) is all real numbers except \( x = 2 \). Therefore, the domain of differentiability is:

\[ \text{Domain} = \mathbb{R} - \{2\} \]

### Final Answer
The domain of differentiability of the function \( f(x) = |x - 2| \cos x \) is \( \mathbb{R} - \{2\} \).

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