The minimum value of the function
f(x) = 2|x - 1| + |x - 2| is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the minimum value of the function \( f(x) = 2|x - 1| + |x - 2| \), we will analyze the function by breaking it down into intervals based on the points where the absolute values change.

### Step 1: Identify the critical points
The critical points occur where the expressions inside the absolute values equal zero. This gives us:
- \( x - 1 = 0 \) → \( x = 1 \)
- \( x - 2 = 0 \) → \( x = 2 \)

Thus, we have three intervals to consider:
1. \( x < 1 \)
2. \( 1 \leq x < 2 \)
3. \( x \geq 2 \)

### Step 2: Analyze the first interval \( x < 1 \)
In this interval, both \( |x - 1| \) and \( |x - 2| \) are negative:
- \( |x - 1| = -(x - 1) = -x + 1 \)
- \( |x - 2| = -(x - 2) = -x + 2 \)

Substituting these into \( f(x) \):
\[ f(x) = 2(-x + 1) + (-x + 2) = -2x + 2 - x + 2 = -3x + 4 \]

### Step 3: Analyze the second interval \( 1 \leq x < 2 \)
In this interval, \( |x - 1| \) is positive and \( |x - 2| \) is negative:
- \( |x - 1| = x - 1 \)
- \( |x - 2| = -(x - 2) = -x + 2 \)

Substituting these into \( f(x) \):
\[ f(x) = 2(x - 1) + (-x + 2) = 2x - 2 - x + 2 = x \]

### Step 4: Analyze the third interval \( x \geq 2 \)
In this interval, both \( |x - 1| \) and \( |x - 2| \) are positive:
- \( |x - 1| = x - 1 \)
- \( |x - 2| = x - 2 \)

Substituting these into \( f(x) \):
\[ f(x) = 2(x - 1) + (x - 2) = 2x - 2 + x - 2 = 3x - 4 \]

### Step 5: Evaluate \( f(x) \) at the critical points
Now we will evaluate \( f(x) \) at the critical points \( x = 1 \) and \( x = 2 \):
- For \( x = 1 \):
\[ f(1) = 2|1 - 1| + |1 - 2| = 2(0) + (1) = 1 \]
- For \( x = 2 \):
\[ f(2) = 2|2 - 1| + |2 - 2| = 2(1) + (0) = 2 \]

### Step 6: Determine the minimum value
Now we compare the values of \( f(x) \) at the critical points:
- \( f(1) = 1 \)
- \( f(2) = 2 \)

Since \( f(x) \) is a linear function in each interval, we can conclude that the minimum value of \( f(x) \) occurs at \( x = 1 \) and is:
\[ \text{Minimum value of } f(x) = 1 \]

### Final Answer
The minimum value of the function \( f(x) = 2|x - 1| + |x - 2| \) is **1**.