If `f(x)=min{|x-1|,|x|,|x+1|,` then the value of `int_-1^1 f(x)dx` is equal to

Options

A. 1
B. `(1)/(2)`
C. `(1)/(4)`
D. `(1)/(8)`

Answer

To solve the problem, we need to calculate the integral of the function \( f(x) = \min\{|x-1|, |x|, |x+1|\} \) over the interval \([-1, 1]\).

### Step 1: Analyze the function \( f(x) \)

The function \( f(x) \) is defined as the minimum of three absolute value functions. We will analyze each component:

1. \( |x-1| \)
2. \( |x| \)
3. \( |x+1| \)

### Step 2: Determine the critical points

The critical points occur where the expressions inside the absolute values equal zero:

- \( |x-1| = 0 \) at \( x = 1 \)
- \( |x| = 0 \) at \( x = 0 \)
- \( |x+1| = 0 \) at \( x = -1 \)

### Step 3: Evaluate \( f(x) \) in different intervals

We will evaluate \( f(x) \) in the intervals defined by the critical points:

1. **For \( x \in [-1, -0.5] \)**:
- \( |x-1| = 1-x \)
- \( |x| = -x \)
- \( |x+1| = x+1 \)
- Thus, \( f(x) = \min\{1-x, -x, x+1\} = x+1 \) (since \( x+1 \) is the smallest value in this interval).

2. **For \( x \in [-0.5, 0] \)**:
- \( |x-1| = 1-x \)
- \( |x| = -x \)
- \( |x+1| = x+1 \)
- Thus, \( f(x) = \min\{1-x, -x, x+1\} = -x \).

3. **For \( x \in [0, 0.5] \)**:
- \( |x-1| = 1-x \)
- \( |x| = x \)
- \( |x+1| = x+1 \)
- Thus, \( f(x) = \min\{1-x, x, x+1\} = x \).

4. **For \( x \in [0.5, 1] \)**:
- \( |x-1| = 1-x \)
- \( |x| = x \)
- \( |x+1| = x+1 \)
- Thus, \( f(x) = \min\{1-x, x, x+1\} = 1-x \).

### Step 4: Set up the integral

Now we can set up the integral as follows:

\[ \int_{-1}^{1} f(x) \, dx = \int_{-1}^{-0.5} (x + 1) \, dx + \int_{-0.5}^{0} (-x) \, dx + \int_{0}^{0.5} x \, dx + \int_{0.5}^{1} (1 - x) \, dx \]

### Step 5: Calculate each integral

1. **Calculate \( \int_{-1}^{-0.5} (x + 1) \, dx \)**:
\[ = \left[ \frac{x^2}{2} + x \right]_{-1}^{-0.5} = \left( \frac{(-0.5)^2}{2} - 0.5 \right) - \left( \frac{(-1)^2}{2} - 1 \right) \]
\[ = \left( \frac{0.25}{2} - 0.5 \right) - \left( \frac{1}{2} - 1 \right) = \left( 0.125 - 0.5 \right) - \left( 0.5 - 1 \right) = -0.375 + 0.5 = 0.125 \]

2. **Calculate \( \int_{-0.5}^{0} (-x) \, dx \)**:
\[ = \left[ -\frac{x^2}{2} \right]_{-0.5}^{0} = 0 - \left( -\frac{(-0.5)^2}{2} \right) = 0.125 \]

3. **Calculate \( \int_{0}^{0.5} x \, dx \)**:
\[ = \left[ \frac{x^2}{2} \right]_{0}^{0.5} = \frac{(0.5)^2}{2} - 0 = \frac{0.25}{2} = 0.125 \]

4. **Calculate \( \int_{0.5}^{1} (1 - x) \, dx \)**:
\[ = \left[ x - \frac{x^2}{2} \right]_{0.5}^{1} = \left( 1 - \frac{1}{2} \right) - \left( 0.5 - \frac{(0.5)^2}{2} \right) \]
\[ = \left( 0.5 \right) - \left( 0.5 - 0.125 \right) = 0.5 - 0.375 = 0.125 \]

### Step 6: Sum the integrals

Now we sum all the calculated integrals:

\[ \int_{-1}^{1} f(x) \, dx = 0.125 + 0.125 + 0.125 + 0.125 = 0.5 \]

### Final Answer

Thus, the value of \( \int_{-1}^{1} f(x) \, dx \) is \( \frac{1}{4} \).