Let `f(x)={(x+1, -1lexle0),(-x,0ltxle1):}` then

Options

A. Intermediate mean value theorem applies to `f(x)` on `[-1,1]`
B. `f(x)` attains maximum and minimum value
C. `f(x)` satisfies Lagrange's mean value theorem on`[-1,1]`
D. `f(x)` is bounded.

Explanation

To solve the problem, we need to analyze the piecewise function given:

\[ f(x) = \begin{cases} x + 1 & \text{for } -1 \leq x \leq 0 \\ -x & \text{for } 0 < x \leq 1 \end{cases} \]

### Step 1: Evaluate the function at the endpoints and check continuity.

1. **At \( x = -1 \)**:
\[ f(-1) = -1 + 1 = 0 \]

2. **At \( x = 0 \)**:
\[ f(0) = 0 + 1 = 1 \]

3. **At \( x = 0 \) from the right** (using the second piece):
\[ f(0^+) = -0 = 0 \]

### Step 2: Check for discontinuity.

The function is discontinuous at \( x = 0 \) because:
- \( f(0) = 1 \) (from the first piece)
- \( f(0^+) = 0 \) (from the second piece)

Since the left-hand limit does not equal the right-hand limit at \( x = 0 \), the function is discontinuous.

### Step 3: Determine the maximum and minimum values.

1. **For \( -1 \leq x \leq 0 \)**:
- The function \( f(x) = x + 1 \) is increasing.
- Minimum at \( x = -1 \): \( f(-1) = 0 \)
- Maximum at \( x = 0 \): \( f(0) = 1 \)

2. **For \( 0 < x \leq 1 \)**:
- The function \( f(x) = -x \) is decreasing.
- Maximum at \( x = 0^+ \): \( f(0^+) = 0 \)
- Minimum at \( x = 1 \): \( f(1) = -1 \)

### Step 4: Combine results to find overall maximum and minimum.

- The overall minimum value of \( f(x) \) is \(-1\) (from \( x = 1 \)).
- The overall maximum value of \( f(x) \) is \(1\) (from \( x = 0 \)).

### Step 5: Check if the function is bounded.

The function is bounded because:
- For \( -1 \leq x \leq 1 \), \( f(x) \) takes values from \(-1\) to \(1\).
- Therefore, \( f(x) \) is bounded in the interval \([-1, 1]\).

### Conclusion

- The function is discontinuous at \( x = 0 \).
- The minimum value is \(-1\) and the maximum value is \(1\).
- The function is bounded.

### Final Answer

- The correct options are:
- Minimum value: \(-1\)
- Maximum value: \(1\)
- The function is bounded.