Define: `f (x) =|x^(2)-4x +3| ln x+2 (x-2)^(1//3) , x gt 0`
`h (x)= {{:(x-1"," , x in Q),(x ^(2) -x-2"," , x cancel (in)Q):}`
`f (x)` is non-differentiable at…… points and the sum of corresponding x value (s) is ……

Options

A. `3,6`
B. `2,3`
C. `2,4`
D. `2,5`

Answer

To solve the problem, we need to analyze the function \( f(x) = |x^2 - 4x + 3| \ln x + 2(x - 2)^{1/3} \) for \( x > 0 \) and determine where it is non-differentiable.

### Step-by-Step Solution:

1. **Identify the components of \( f(x) \)**:
- The first part is \( |x^2 - 4x + 3| \).
- The second part is \( 2(x - 2)^{1/3} \).

2. **Factor the quadratic expression**:
- The expression \( x^2 - 4x + 3 \) can be factored as:
\[ x^2 - 4x + 3 = (x - 1)(x - 3) \]
- Thus, we can rewrite \( f(x) \) as:
\[ f(x) = |(x - 1)(x - 3)| \ln x + 2(x - 2)^{1/3} \]

3. **Determine points of non-differentiability**:
- The absolute value function \( |(x - 1)(x - 3)| \) introduces potential non-differentiability at \( x = 1 \) and \( x = 3 \).
- The term \( 2(x - 2)^{1/3} \) is non-differentiable at \( x = 2 \) because the derivative of \( x^{1/3} \) is undefined at \( x = 0 \).

4. **Check differentiability at the critical points**:
- **At \( x = 1 \)**:
- Left-hand derivative (LHD) and right-hand derivative (RHD) need to be checked.
- For \( x < 1 \), \( f(x) = -(x - 1)(x - 3) \ln x \).
- For \( x > 1 \), \( f(x) = (x - 1)(x - 3) \ln x \).
- Both derivatives yield \( 0 \) at \( x = 1 \), hence \( f(x) \) is differentiable at \( x = 1 \).

- **At \( x = 2 \)**:
- The derivative of \( 2(x - 2)^{1/3} \) is not defined at \( x = 2 \).
- Thus, \( f(x) \) is non-differentiable at \( x = 2 \).

- **At \( x = 3 \)**:
- For \( x < 3 \), \( f(x) = -(x - 1)(x - 3) \ln x \).
- For \( x > 3 \), \( f(x) = (x - 1)(x - 3) \ln x \).
- The left-hand and right-hand derivatives at \( x = 3 \) yield different values, indicating non-differentiability at \( x = 3 \).

5. **Conclusion**:
- The points where \( f(x) \) is non-differentiable are \( x = 2 \) and \( x = 3 \).
- The sum of these points is:
\[ 2 + 3 = 5 \]

### Final Answer:
- \( f(x) \) is non-differentiable at **2 points** and the sum of corresponding \( x \) values is **5**.