To solve the problem, we need to analyze the function \( f(x) = |x - \frac{1}{2}| + |x - 1| + \tan x \) and determine the points in the interval \([0, 2]\) where it is not differentiable.
### Step 1: Identify the components of the function
The function \( f(x) \) consists of three parts:
1. \( g(x) = |x - \frac{1}{2}| \)
2. \( h(x) = |x - 1| \)
3. \( k(x) = \tan x \)
### Step 2: Analyze \( g(x) = |x - \frac{1}{2}| \)
The function \( g(x) \) is not differentiable at the point where the expression inside the absolute value equals zero:
\[
x - \frac{1}{2} = 0 \implies x = \frac{1}{2}
\]
Thus, \( g(x) \) is not differentiable at \( x = \frac{1}{2} \).
### Step 3: Analyze \( h(x) = |x - 1| \)
Similarly, for \( h(x) \):
\[
x - 1 = 0 \implies x = 1
\]
Thus, \( h(x) \) is not differentiable at \( x = 1 \).
### Step 4: Analyze \( k(x) = \tan x \)
The function \( k(x) \) is not differentiable where it is undefined. The tangent function is undefined at:
\[
x = \frac{\pi}{2} \approx 1.57
\]
Since \( \frac{\pi}{2} \) lies within the interval \([0, 2]\), \( k(x) \) is not differentiable at \( x = \frac{\pi}{2} \).
### Step 5: Compile the points of non-differentiability
From the analysis, we have identified the following points where \( f(x) \) is not differentiable:
1. \( x = \frac{1}{2} \) (from \( g(x) \))
2. \( x = 1 \) (from \( h(x) \))
3. \( x = \frac{\pi}{2} \) (from \( k(x) \))
### Step 6: Count the points in the interval \([0, 2]\)
Now we need to check which of these points lie within the interval \([0, 2]\):
- \( \frac{1}{2} \) is in \([0, 2]\)
- \( 1 \) is in \([0, 2]\)
- \( \frac{\pi}{2} \approx 1.57 \) is also in \([0, 2]\)
### Conclusion
All three points \( \frac{1}{2}, 1, \text{ and } \frac{\pi}{2} \) are within the interval \([0, 2]\). Therefore, the number of points where \( f(x) \) is not differentiable in the interval \([0, 2]\) is 3.
Thus, the answer is:
**(c) 3**
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