Question

The value of \( c \) in Rolle's Theorem for the function \[ f(x) = e^x \sin x, \, x \in [0, \pi] \] is:

• A. \( \frac{\pi}{6} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{2} \)
• D. \( \frac{3\pi}{4} \)

Hint:

Rolle's Theorem is useful when the function is continuous and differentiable in the interval, and the function takes the same value at the endpoints.

Answer 1

The Correct Option is C

Step 1: Rolle's Theorem guarantees that there exists at least one point \( c \) in the interval \( [0, \pi] \) where the derivative of \( f(x) \) is zero.
Step 2: To find \( c \), first compute the derivative of \( f(x) = e^x \sin x \). We get: \[ f'(x) = e^x (\sin x + \cos x). \] Set \( f'(x) = 0 \), which gives \( \sin x + \cos x = 0 \). Solving this gives \( x = \frac{\pi}{2} \).

Final Answer: \[ \boxed{\frac{\pi}{2}} \]