If `f(x) = 4^(sin x)` satisfies the Rolle's theorem on `[0, pi]`, then the value of `c in (0, pi)` for which f'(c ) = 0 is

Options

A. `c = (pi)/(6)`
B. `c = (pi)/(4)`
C. `c= (pi)/(2)`
D. `c= (pi)/(3)`

Answer

To solve the problem, we need to find the value of \( c \) in the interval \( (0, \pi) \) such that \( f'(c) = 0 \) for the function \( f(x) = 4^{\sin x} \). We will also verify that the function satisfies the conditions of Rolle's Theorem.

### Step 1: Verify the conditions of Rolle's Theorem
Rolle's Theorem states that if a function is continuous on a closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \( f(a) = f(b) \), then there exists at least one \( c \) in \((a, b)\) such that \( f'(c) = 0 \).

1. **Check continuity and differentiability**:
- The function \( f(x) = 4^{\sin x} \) is continuous and differentiable for all \( x \) since it is composed of continuous and differentiable functions (exponential and sine functions).

2. **Check endpoints**:
- Calculate \( f(0) \) and \( f(\pi) \):
\[ f(0) = 4^{\sin(0)} = 4^0 = 1 \]
\[ f(\pi) = 4^{\sin(\pi)} = 4^0 = 1 \]
- Since \( f(0) = f(\pi) = 1 \), the conditions of Rolle's Theorem are satisfied.

### Step 2: Find the derivative \( f'(x) \)
To find \( c \) such that \( f'(c) = 0 \), we first need to compute the derivative \( f'(x) \).

Using the chain rule:
\[ f(x) = 4^{\sin x} = e^{\sin x \ln 4} \]
Now, differentiate using the chain rule:
\[ f'(x) = \frac{d}{dx}(e^{\sin x \ln 4}) = e^{\sin x \ln 4} \cdot \frac{d}{dx}(\sin x \ln 4) \]
\[ = e^{\sin x \ln 4} \cdot \ln 4 \cdot \cos x \]
So,
\[ f'(x) = 4^{\sin x} \cdot \ln 4 \cdot \cos x \]

### Step 3: Set the derivative to zero
Now we need to find \( c \) such that:
\[ f'(c) = 0 \]
This implies:
\[ 4^{\sin c} \cdot \ln 4 \cdot \cos c = 0 \]
Since \( 4^{\sin c} \) and \( \ln 4 \) are never zero, we have:
\[ \cos c = 0 \]

### Step 4: Solve for \( c \)
The cosine function is zero at:
\[ c = \frac{\pi}{2} + n\pi \]
For \( c \) to be in the interval \( (0, \pi) \), we take \( n = 0 \):
\[ c = \frac{\pi}{2} \]

### Conclusion
Thus, the value of \( c \) in the interval \( (0, \pi) \) for which \( f'(c) = 0 \) is:
\[ \boxed{\frac{\pi}{2}} \]