Consider `f(x)=|1-x|,1 le xle2 and g(x)=f(x)+b sin.(pi)/(2)x, 1 le xle 2` then which of the following is correct?

Options

A. Rolle's theorem is applicable to both f and g with `b=(3)/(2)`.
B. LMVT is not applicable to f and Rolle's theorem is applicable to g with `b=(1)/(2)`
C. LMVT is applicable to f and Rolle's theorem is applicable to g with b = 1.
D. Rolle's theorem is not applicable to both f and g for any real b.

Explanation

To solve the problem, we will analyze the functions \( f(x) \) and \( g(x) \) step by step.

### Step 1: Define the function \( f(x) \)
The function is given as:
\[ f(x) = |1 - x| \]
For \( 1 \leq x \leq 2 \), we can simplify \( f(x) \):
- When \( x = 1 \), \( f(1) = |1 - 1| = 0 \)
- When \( x = 2 \), \( f(2) = |1 - 2| = 1 \)

Since \( 1 \leq x \leq 2 \), we can express \( f(x) \) as:
\[ f(x) = x - 1 \quad \text{for } 1 \leq x \leq 2 \]

### Step 2: Define the function \( g(x) \)
The function \( g(x) \) is defined as:
\[ g(x) = f(x) + b \sin\left(\frac{\pi}{2} x\right) \]
Substituting \( f(x) \):
\[ g(x) = (x - 1) + b \sin\left(\frac{\pi}{2} x\right) \]

### Step 3: Evaluate \( g(x) \) at the endpoints
Now we will evaluate \( g(x) \) at the endpoints \( x = 1 \) and \( x = 2 \):
- For \( x = 1 \):
\[ g(1) = (1 - 1) + b \sin\left(\frac{\pi}{2} \cdot 1\right) = 0 + b \cdot 1 = b \]
- For \( x = 2 \):
\[ g(2) = (2 - 1) + b \sin\left(\frac{\pi}{2} \cdot 2\right) = 1 + b \cdot 0 = 1 \]

### Step 4: Apply Rolle's Theorem
For Rolle's Theorem to be applicable, we need \( g(1) = g(2) \):
\[ b = 1 \]

### Step 5: Check the conditions for Mean Value Theorem (MVT)
- \( f(x) \) is continuous on the closed interval \([1, 2]\) and differentiable on the open interval \( (1, 2) \).
- \( g(x) \) is continuous on \([1, 2]\) and differentiable on \( (1, 2) \) when \( b = 1 \).

### Conclusion
- Since \( f(1) \neq f(2) \), Rolle's Theorem is not applicable to \( f(x) \).
- However, Mean Value Theorem is applicable to \( f(x) \) and Rolle's Theorem is applicable to \( g(x) \) when \( b = 1 \).

Thus, the correct option is:
**Mean Value Theorem is applicable to \( f \) and Rolle's Theorem is applicable to \( g \).**