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) = x (x - 1)² ; [0, 1]

Verified step by step guidance

1. First, understand Rolle's Theorem: It states that if a function f(x) 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.
2. Check the continuity of the function ƒ(x) = x(x - 1)² on the interval [0, 1]. Since ƒ(x) is a polynomial, it is continuous everywhere, including on [0, 1].
3. Verify differentiability of ƒ(x) on the open interval (0, 1). Again, since ƒ(x) is a polynomial, it is differentiable everywhere, including on (0, 1).
4. Evaluate ƒ(x) at the endpoints of the interval: Calculate ƒ(0) and ƒ(1). If ƒ(0) = ƒ(1), then the function satisfies the condition f(a) = f(b) required by Rolle's Theorem.
5. Find the derivative ƒ'(x) and solve ƒ'(x) = 0 to find the point(s) c in the interval (0, 1) where the derivative is zero. This will give the point(s) guaranteed by Rolle's Theorem.

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

A function is continuous on an interval if there are no breaks, jumps, or holes in its graph. For Rolle's Theorem to apply, the function must be continuous on the closed interval [a, b]. This means that the function must be defined at every point in the interval and the limit of the function as it approaches any point within the interval must equal the function's value at that point.

Differentiability

A function is differentiable on an interval if it has a derivative at every point in that interval. For Rolle's Theorem to hold, the function must be differentiable on the open interval (a, b). This implies that the function must not have any sharp corners or vertical tangents within the interval, ensuring that the slope can be calculated at all points.