`f(x) = |x|+|x-1| +|x-2|`, then which one of the following is not correct ?

Options

A. f(x) has a minimum at x=1
B. f(x) has a maximum at x=0
C. f(x) has niether a maximum nor a minimum at x=0
D. f(x) has niether a maximum nor a minimum x=2

Answer

To solve the problem \( f(x) = |x| + |x-1| + |x-2| \) and determine which statement is not correct, we will analyze the function step by step.

### Step 1: Identify the critical points
The absolute value function changes at the points where the expression inside the absolute value equals zero. For \( f(x) \), the critical points are:
- \( x = 0 \) (where \( |x| \) changes)
- \( x = 1 \) (where \( |x-1| \) changes)
- \( x = 2 \) (where \( |x-2| \) changes)

### Step 2: Analyze the function in different intervals
We will analyze \( f(x) \) in the intervals determined by the critical points:
1. \( x < 0 \)
2. \( 0 \leq x < 1 \)
3. \( 1 \leq x < 2 \)
4. \( x \geq 2 \)

#### Interval 1: \( x < 0 \)
In this interval, all terms are negative:
\[ f(x) = -x + (1 - x) + (2 - x) = 3 - 3x \]

#### Interval 2: \( 0 \leq x < 1 \)
In this interval, \( |x| \) is positive while the other two terms are still positive:
\[ f(x) = x + (1 - x) + (2 - x) = 3 - x \]

#### Interval 3: \( 1 \leq x < 2 \)
In this interval, \( |x| \) and \( |x-1| \) are positive while \( |x-2| \) is negative:
\[ f(x) = x + (x - 1) + (2 - x) = x + 1 \]

#### Interval 4: \( x \geq 2 \)
In this interval, all terms are positive:
\[ f(x) = x + (x - 1) + (x - 2) = 3x - 3 \]

### Step 3: Determine the values at critical points
Now we will find the values of \( f(x) \) at the critical points:
- At \( x = 0 \):
\[ f(0) = |0| + |0-1| + |0-2| = 0 + 1 + 2 = 3 \]
- At \( x = 1 \):
\[ f(1) = |1| + |1-1| + |1-2| = 1 + 0 + 1 = 2 \]
- At \( x = 2 \):
\[ f(2) = |2| + |2-1| + |2-2| = 2 + 1 + 0 = 3 \]

### Step 4: Analyze the behavior of \( f(x) \)
- For \( x < 0 \), \( f(x) = 3 - 3x \) is increasing.
- For \( 0 \leq x < 1 \), \( f(x) = 3 - x \) is decreasing.
- For \( 1 \leq x < 2 \), \( f(x) = x + 1 \) is increasing.
- For \( x \geq 2 \), \( f(x) = 3x - 3 \) is increasing.

### Step 5: Identify minimum and maximum points
- The minimum value occurs at \( x = 1 \) where \( f(1) = 2 \).
- The function does not have a maximum value since it increases indefinitely for \( x \geq 2 \).

### Conclusion
From the analysis:
- \( f(x) \) has a minimum at \( x = 1 \).
- \( f(x) \) does not have a maximum value.
- At \( x = 2 \), it neither has a maximum nor a minimum.

### Final Answer
The statement that is not correct is: "f(x) has a maximum at \( x = 0 \)".