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) = 1 - | x | ; [-1, 1]

Verified step by step guidance

1. First, recall Rolle's Theorem, which states that if a function f 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 f(x) = 1 - |x| on the interval [-1, 1]. The function is continuous on [-1, 1] because the absolute value function is continuous everywhere.
3. Verify that f is differentiable on the open interval (-1, 1). The function f(x) = 1 - |x| is differentiable on (-1, 1) except possibly at x = 0, where the absolute value function has a corner.
4. Check the condition f(-1) = f(1). Calculate f(-1) = 1 - |-1| = 0 and f(1) = 1 - |1| = 0. Since f(-1) = f(1), this condition is satisfied.
5. Since f is not differentiable at x = 0, Rolle's Theorem does not apply to this function on the interval [-1, 1].

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 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, indicating potential maxima, minima, or points of inflection.

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's value at the endpoints must be defined and the limit as x approaches any point in 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. Differentiability implies continuity, but not vice versa. For Rolle's Theorem to hold, the function must be differentiable on the open interval (a, b), meaning it should not have any sharp corners or vertical tangents within that range.