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 guarantees the existence of at least one point where the derivative is zero if the function is continuous and differentiable on the interval, and \( f(a) = f(b) \).

Answer 1

The Correct Option is D

According to Rolle’s Theorem, there exists at least one point \( c \) in \( (a, b) \) such that \( f'(c) = 0 \), where \( f(a) = f(b) \). For the given function, we find that \( c = \frac{3\pi}{4} \) satisfies the condition.