11–18. Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle’s Theorem.




ƒ(x) = sin 2x; [0, π/2]

Verified step by step guidance

1. First, recall Rolle's Theorem, which states that if a function f 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 (a, b) such that f'(c) = 0.
2. Check the function f(x) = sin(2x) for continuity on the interval [0, π/2]. Since sine is a continuous function and 2x is a linear transformation, f(x) is continuous on [0, π/2].
3. Verify that f(x) is differentiable on the open interval (0, π/2). The function sin(2x) is differentiable everywhere, so it is differentiable on (0, π/2).
4. Evaluate f at the endpoints of the interval: f(0) = sin(2*0) = sin(0) = 0 and f(π/2) = sin(2*(π/2)) = sin(π) = 0. Since f(0) = f(π/2), the condition f(a) = f(b) is satisfied.
5. Find the derivative f'(x) = d/dx [sin(2x)] = 2cos(2x). Set f'(x) = 0 to find the point(s) guaranteed by Rolle's Theorem: 2cos(2x) = 0. Solve for x in the interval (0, π/2) to find the value(s) of c where f'(c) = 0.

Key Concepts

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 the function values at the endpoints are equal (f(a) = f(b)), then there exists at least one c in (a, b) such that f'(c) = 0. This theorem is essential for finding critical points where the function's slope is zero.

Continuity and Differentiability

For Rolle's Theorem to apply, the function must be continuous on the closed interval and differentiable on the open interval. Continuity ensures there are no breaks or jumps in the function, while differentiability means the function has a defined derivative at every point in the interval, allowing for the application of the theorem.

Finding Critical Points

Once it is established that Rolle's Theorem applies, the next step is to find the critical points where the derivative of the function equals zero. This involves calculating the derivative of the function and solving for x to find the point(s) c in the interval (a, b) where the slope of the tangent line is horizontal, indicating a local extremum.