For which of the following function 9s) Lagrange's mean value theorem is not applicable in `[1,2]` ?

Options

A. `f (x)={{:((3)/(2)-x "," , x lt 3/2),(((3)/(2)-x)^(2)"," , x ge 3/2):}`
B. `f (x)={{:((sin (x-1))/(x-1)"," , x ne 1),( 1 "," , x =1):}`
C. `f (x)=(x-1) |x+1|`
D. `f (x) =|x-1|`

Answer

To determine for which function Lagrange's Mean Value Theorem (LMVT) is not applicable in the interval [1, 2], we need to check the continuity and differentiability of each function in that interval. The theorem states that a function must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b) for LMVT to be applicable.

### Step-by-step Solution:

1. **Identify the Functions**: Let's denote the functions given in the options as follows:
- Option A: \( f(x) = \begin{cases} 3 - \frac{3}{2 - x} & \text{for } x < \frac{3}{2} \\ \frac{3}{2 - x}^2 & \text{for } x \geq \frac{3}{2} \end{cases} \)
- Option B: \( f(x) = \frac{\sin(x) - 1}{x - 1} \) for \( x \neq 1 \) and \( f(1) = 0 \)
- Option C: \( f(x) = (x - 1)|x + 1| \)
- Option D: \( f(x) = |x - 1| \)

2. **Check Continuity for Option A**:
- We need to check the continuity at \( x = \frac{3}{2} \).
- Calculate \( \lim_{x \to \frac{3}{2}^-} f(x) \) and \( \lim_{x \to \frac{3}{2}^+} f(x) \):
- \( \lim_{x \to \frac{3}{2}^-} f(x) = 3 - \frac{3}{2 - \frac{3}{2}} = 0 \)
- \( \lim_{x \to \frac{3}{2}^+} f(x) = \left(\frac{3}{2 - \frac{3}{2}}\right)^2 = 0 \)
- Since both limits are equal, \( f(x) \) is continuous at \( x = \frac{3}{2} \).

3. **Check Differentiability for Option A**:
- We need to find the left-hand derivative and right-hand derivative at \( x = \frac{3}{2} \):
- Left-hand derivative:
\[ f'(x) = \frac{d}{dx}(3 - \frac{3}{2 - x}) \Rightarrow f'(\frac{3}{2}^-) = -1 \]
- Right-hand derivative:
\[ f'(x) = \frac{d}{dx}(\frac{3}{2 - x}^2) \Rightarrow f'(\frac{3}{2}^+) = 0 \]
- Since the left-hand derivative does not equal the right-hand derivative, \( f(x) \) is not differentiable at \( x = \frac{3}{2} \).

4. **Conclusion for Option A**:
- Since \( f(x) \) is continuous but not differentiable at \( x = \frac{3}{2} \), Lagrange's Mean Value Theorem is not applicable for Option A.

5. **Check Other Options**:
- **Option B**: \( f(x) \) is continuous and differentiable in [1, 2].
- **Option C**: \( f(x) \) is continuous and differentiable in [1, 2].
- **Option D**: \( f(x) \) is continuous and differentiable in [1, 2].

### Final Answer:
Lagrange's Mean Value Theorem is not applicable for **Option A**.