Rolle's Theorem Exercises
Example 1
Let f(x) = sin(x). This function is differentiable everywhere. Prove that there is some c in (0, 2π) with f ' (c) = 0. By looking at the graph of f, determine how many such values of c there are in (0, 2π).
Example 2
For the function f(x) = 2^{x}, determine whether we're allowed to use Rolle's Theorem to guarantee the existence of some c in (0, 1) with f ' (c) = 0. If not, explain why not.
Example 3
For the function f shown below, determine whether we're allowed to use Rolle's Theorem to guarantee the existence of some c in (-1 ,1) with f ' (c) = 0. If not, explain why not.
(Insert graph of f(x) = 3 for x ≤ -1, f(x) = x^{2} for -1 < x < 1 and f(x) = 3 for x ≥ 1)
Example 4
For the function f shown below, determine we're allowed to use Rolle's Theorem to guarantee the existence of some c in (a, b) with f ' (c) = 0. If not, explain why not.
(Insert graph of the function f(x) = -2(x-a) for x ≤ a, f(x) = 0 for a < x < b and f(x) = 2(x-b) for x ≥ b)
Example 5
For the function f shown below, determine we're allowed to use Rolle's Theorem to guarantee the existence of some c in (a, b) with f ' (c) = 0. If not, explain why not.
(Insert graph of f(x) = sin(x) on the interval (0, 2π) On the x-axis, label the origin as a, and then label x = 3π/2 as b.)
Example 6
For each given function f and interval, determine if Rolle's Theorem guarantees the existence of some c in that interval with f ' (c) = 0. If not, explain why not.
- f(x) = x^{3} on the interval (-2, 2)
Example 7
For the function f and interval, determine if Rolle's Theorem guarantees the existence of some c in that interval with f ' (c) = 0. If not, explain why not.
- f(x) = cos(x) on the interval (-π, 3π) (yes, cos(x) is differentiable)
Example 8
For the given function f and interval, determine if Rolle's Theorem guarantees the existence of some c in that interval with f ' (c) = 0. If not, explain why not.
- f(x) = (x – 2)^{2} + 4 on the interval (-2, 2)
Example 9
For the given function f and interval, determine if Rolle's Theorem guarantees the existence of some c in that interval with f ' (c) = 0. If not, explain why not.
- on the interval (-1, 1).