a.
Apply Rolle's theorem to f(x) = sinx√cos2x in (0,) and find c.
b
Evaluate lim logtanx tan2x
x-0

Let's go through each part of the problem step by step.

### Part a: Apply Rolle's Theorem to \( f(x) = \sin(x) \sqrt{\cos(2x)} \) on the interval \( (0, \pi) \).

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

#### Step 1: Check the conditions of Rolle's Theorem.

1. **Continuity**: The function \( f(x) = \sin(x) \sqrt{\cos(2x)} \

Learn how to apply Rolle's Theorem to the function f(x) = sin(x) sqrt(cos(2x)) on the interval (0, π). This problem involves differentiation and the application of Rolle's Theorem, providing insights into the conditions for which the theorem can be applied. Suitable for college-level students studying calculus.

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