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. Check if the function ƒ(x) = 1 - |x| is continuous on the closed interval [-1, 1]. Since it is a piecewise linear function, it is continuous everywhere, including at the endpoints of the interval.
2. Verify if the function is differentiable on the open interval (-1, 1). The function is differentiable everywhere except at x = 0, where the absolute value function has a corner.
3. Evaluate the function at the endpoints of the interval: calculate ƒ(-1) and ƒ(1) to check if they are equal. This is necessary for Rolle's Theorem to apply.
4. Since ƒ(-1) = ƒ(1), we can conclude that the conditions for Rolle's Theorem are satisfied on the interval [-1, 1] except at the point x = 0.
5. To find the point(s) guaranteed by Rolle's Theorem, compute the derivative of the function and set it equal to zero in the interval (-1, 1) to find critical points.