Let `f (x) =||x^(2)-4x+3|-2|.` Which of the following is/are correct ?

Options

A. `f (x)=` m has exactly two real solutios of different sign `AA m gt2`
B. `f (x)=m` has exactly two real solution `AA m in (2,oo) uu{0}`
C. `f (x)=m` has no solutions `AA m lt 0`
D. `f(x)=m` has four distinct real solution `AA m in (0,1)`

Answer

To solve the problem, we need to analyze the function \( f(x) = ||x^2 - 4x + 3| - 2| \) and determine the validity of the given statements regarding the solutions of the equation \( f(x) = m \).

### Step 1: Simplify the inner expression
First, we simplify the expression inside the outer absolute value.

1. The quadratic \( x^2 - 4x + 3 \) can be factored as:
\[ x^2 - 4x + 3 = (x - 1)(x - 3) \]
This quadratic has roots at \( x = 1 \) and \( x = 3 \).

### Step 2: Analyze the quadratic
2. The quadratic opens upwards (since the coefficient of \( x^2 \) is positive) and is zero at \( x = 1 \) and \( x = 3 \). The vertex of the parabola occurs at \( x = 2 \), where:
\[ f(2) = 2^2 - 4 \cdot 2 + 3 = 1 \]
Thus, the minimum value of \( |x^2 - 4x + 3| \) is 0 (at \( x = 1 \) and \( x = 3 \)) and the maximum value occurs at the vertex.

### Step 3: Determine the range of the inner absolute value
3. The maximum value of \( |x^2 - 4x + 3| \) occurs at the vertex \( x = 2 \):
\[ |1| = 1 \]
Therefore, \( |x^2 - 4x + 3| \) ranges from 0 to \( \infty \).

### Step 4: Analyze the outer absolute value
4. Now we analyze \( ||x^2 - 4x + 3| - 2| \):
- For \( |x^2 - 4x + 3| < 2 \) (which occurs between the roots), the expression will be negative, so we take the negative:
\[ ||x^2 - 4x + 3| - 2| = 2 - |x^2 - 4x + 3| \]
- For \( |x^2 - 4x + 3| \geq 2 \), the expression remains positive:
\[ ||x^2 - 4x + 3| - 2| = |x^2 - 4x + 3| - 2 \]

### Step 5: Determine the critical points
5. The critical points occur at \( x = 1 \) and \( x = 3 \), where \( |x^2 - 4x + 3| = 0 \). The function \( f(x) \) will equal 2 at points where the inner expression equals 2.

### Step 6: Analyze the statements
Now we analyze the statements based on the graph and the behavior of \( f(x) \):

1. **First Statement**: \( f(x) = m \) has exactly two real solutions of different signs for all \( m > 2 \).
- **Correct**: The graph of \( f(x) \) will intersect the horizontal line \( y = m \) at two points.

2. **Second Statement**: \( f(x) = m \) has exactly two real solutions for all \( m \) in \( [0, 2) \).
- **Correct**: The function will cross the line \( y = m \) twice in this range.

3. **Third Statement**: \( f(x) = m \) has no solution for all \( m < 0 \).
- **Correct**: Since \( f(x) \) is always non-negative, there are no solutions for negative \( m \).

4. **Fourth Statement**: \( f(x) = m \) has exactly four distinct solutions for all \( m \) in \( (0, 1) \).
- **Correct**: The graph will intersect the horizontal line \( y = m \) four times in this interval.

### Conclusion
All statements are correct.