The value of c is Rolle 's theorem for the function `f(x)=e^(x)sinx,x in[0,pi]` , is

Options

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

Answer

To solve the problem using Rolle's theorem for the function \( f(x) = e^x \sin x \) on the interval \( [0, \pi] \), we will follow these steps:

### Step 1: Verify the conditions of Rolle's theorem
Rolle's theorem states that if a function \( f \) 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. **Continuity**: The function \( f(x) = e^x \sin x \) is a product of two continuous functions (exponential and sine), hence it is continuous on \([0, \pi]\).
2. **Differentiability**: The function is also differentiable on the open interval \((0, \pi)\) because both \( e^x \) and \( \sin x \) are differentiable everywhere.
3. **Equal values at endpoints**: We need to check if \( f(0) = f(\pi) \):
- \( f(0) = e^0 \sin(0) = 1 \cdot 0 = 0 \)
- \( f(\pi) = e^\pi \sin(\pi) = e^\pi \cdot 0 = 0 \)
- Since \( f(0) = f(\pi) = 0 \), the conditions of Rolle's theorem are satisfied.

### Step 2: Find the derivative of the function
Next, we find the derivative \( f'(x) \):
\[ f'(x) = \frac{d}{dx}(e^x \sin x) = e^x \sin x + e^x \cos x = e^x (\sin x + \cos x) \]

### Step 3: Set the derivative equal to zero
According to Rolle's theorem, we set the derivative equal to zero to find \( c \):
\[ f'(c) = e^c (\sin c + \cos c) = 0 \]
Since \( e^c \) is never zero, we can simplify this to:
\[ \sin c + \cos c = 0 \]

### Step 4: Solve for \( c \)
Rearranging the equation gives:
\[ \sin c = -\cos c \]
Dividing both sides by \( \cos c \) (where \( \cos c \neq 0 \)):
\[ \tan c = -1 \]
The general solution for \( \tan c = -1 \) is:
\[ c = \frac{3\pi}{4} + n\pi \quad (n \in \mathbb{Z}) \]
Since we are looking for \( c \) in the interval \( (0, \pi) \), we take:
\[ c = \frac{3\pi}{4} \]

### Conclusion
Thus, the value of \( c \) that satisfies Rolle's theorem for the function \( f(x) = e^x \sin x \) on the interval \([0, \pi]\) is:
\[ \boxed{\frac{3\pi}{4}} \]