`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, we need to analyze the function \( f(x) = |x| + |x-1| + |x-2| \) and determine the intervals where it is defined and its behavior in those intervals. We will also identify which of the given options is incorrect.

### Step 1: Identify the critical points
The critical points of the function occur where the expressions inside the absolute values change sign. These points are:
- \( x = 0 \)
- \( x = 1 \)
- \( x = 2 \)

### Step 2: Analyze the function in different intervals
We will break down the function into intervals based on the critical points:

1. **Interval 1: \( x < 0 \)**
- Here, \( |x| = -x \), \( |x-1| = -(x-1) = -x + 1 \), and \( |x-2| = -(x-2) = -x + 2 \).
- Thus,
\[ f(x) = -x + (-x + 1) + (-x + 2) = -3x + 3. \]

2. **Interval 2: \( 0 \leq x < 1 \)**
- Here, \( |x| = x \), \( |x-1| = -(x-1) = -x + 1 \), and \( |x-2| = -(x-2) = -x + 2 \).
- Thus,
\[ f(x) = x + (-x + 1) + (-x + 2) = 3 - x. \]

3. **Interval 3: \( 1 \leq x < 2 \)**
- Here, \( |x| = x \), \( |x-1| = x - 1 \), and \( |x-2| = -(x-2) = -x + 2 \).
- Thus,
\[ f(x) = x + (x - 1) + (-x + 2) = x + 1. \]

4. **Interval 4: \( x \geq 2 \)**
- Here, \( |x| = x \), \( |x-1| = x - 1 \), and \( |x-2| = x - 2 \).
- Thus,
\[ f(x) = x + (x - 1) + (x - 2) = 3x - 3. \]

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

### Step 4: Analyze the behavior of the function
Now we can summarize the behavior of \( f(x) \):
- For \( x < 0 \): \( f(x) = -3x + 3 \) (decreasing)
- At \( x = 0 \): \( f(0) = 3 \)
- For \( 0 \leq x < 1 \): \( f(x) = 3 - x \) (decreasing)
- At \( x = 1 \): \( f(1) = 1 \) (minimum point)
- For \( 1 \leq x < 2 \): \( f(x) = x + 1 \) (increasing)
- At \( x = 2 \): \( f(2) = 3 \)
- For \( x \geq 2 \): \( f(x) = 3x - 3 \) (increasing)

### Conclusion
From our analysis, we can conclude:
- \( f(x) \) has a minimum at \( x = 1 \).
- \( f(x) \) does not have a minimum or maximum at \( x = 0 \) since it is decreasing to that point and then decreases again after it.

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