Find the which function does not obey mean value theorem in `[0, 1]`

Options

A. `f(x)=|x|`
B. `f(x)=x|x|`
C. `f(x)={{:((sinx)/(x)",",xne0),(1",",x=0):}`
D. none of these

Answer

To determine which function does not obey the Mean Value Theorem (MVT) in the interval \([0, 1]\), we need to check the conditions of the MVT for each function provided. The MVT states that if a function \( f(x) \) is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one point \( c \) in \((a, b)\) such that:

\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]

### Step-by-Step Solution:

1. **Understand the Conditions of MVT**:
- The function must be continuous on the closed interval \([0, 1]\).
- The function must be differentiable on the open interval \((0, 1)\).

2. **Check Each Function**:
- **Option 1: \( f(x) = |x| \)**
- **Continuity**: The function \( |x| \) is continuous on \([0, 1]\).
- **Differentiability**: The function is differentiable in \((0, 1)\) but not at \( x = 0 \) (it has a sharp corner).
- **Conclusion**: This function does not satisfy the differentiability condition at \( x = 0 \).

- **Option 2: \( f(x) = x \cdot |x| \)**
- **Continuity**: The function \( x \cdot |x| \) is continuous on \([0, 1]\).
- **Differentiability**: The function is differentiable on \((0, 1)\) since it is a polynomial in that interval.
- **Conclusion**: This function satisfies both conditions of MVT.

- **Option 3: \( f(x) = \frac{\sin x}{x} \) for \( x \neq 0 \) and \( f(0) = 1 \)**
- **Continuity**: The function is continuous at \( x = 0 \) since \(\lim_{x \to 0} \frac{\sin x}{x} = 1\).
- **Differentiability**: The function is differentiable at \( x = 0 \) since the derivative exists and is defined.
- **Conclusion**: This function satisfies both conditions of MVT.

- **Option 4: None of the above**
- This option is not applicable since we have identified a function that does not satisfy the MVT.

3. **Final Conclusion**:
- The function that does not obey the Mean Value Theorem in the interval \([0, 1]\) is **Option 1: \( f(x) = |x| \)**.