Rolle's Theorem says:
Let f be a function that
- is continuous on the closed interval [a, b]
- is differentiable on the open interval (a, b), and
- has f (a) = f (b).
Then there is some c in the open interval (a, b) with f ' (c) = 0.
Sometimes the third condition is stated as f (a) = f (b) = 0, but for the proof, it doesn't matter.
In pictures, we're saying suppose f is a nice smooth function with the same starting and ending height:
If f increases or decreases from its starting height, it needs to turn around and come back in order to end at the same height it started at:
Since f is a nice smooth differentiable function, its derivative at that turn-around point must be 0:
If f doesn't go up or down from its starting point, then f is constant:
In this case, f ' (c) is 0 for every value of c in the interval (a, b).
Rolle's Theorem is reminiscent of the Intermediate Value Theorem. Rolle's Theorem says if f satisfies some assumptions (more mathematically known as hypotheses) then f ' will be zero at some point in (a, b). We could have a constant function, in which case f ' will be 0 infinitely many times:
We could have a function that turns around once:
Or we could have a function that turns around many times:
Rolle's Theorem doesn't tell us where or how many times f ' will be zero; it tells us f ' must be zero at least once if the hypotheses are all satisfied.
Sample Problem
Suppose f is not continuous on [a, b]. Then there doesn't need to be any c in (a, b) with f ' (c) = 0. Here's an example:
This function is not continuous. At the point of discontinuity, f ' doesn't exist. At all other points in the interval, f ' is positive:
There is no point c in (a, b) where f ' (c) = 0.
We found earlier that the derivative of the absolute value function doesn't exist at 0. When x is negative the slope of the absolute value function is -1; when x is positive the slope of the absolute value function is 1:
There is no value of c anywhere, in any interval (a, b), with f ' (c) = 0. The derivative of the absolute value function isn't 0 anywhere.
If a function fails any of the hypotheses, we aren't allowed to use Rolle's Theorem.
Example 1
Let f(x) = x2. Prove that there is some c in (-2, 2) with f ' (c) = 0. |
Example 2
For the function f shown below, determine if 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.
|
Example 3
Let |
. Determine if Rolle's Theorem guarantees the existence of some c in (-1, 1) with f ' (c) = 0. If not, explain why not.
Example 4
Let f(x) = x2 – x. Does Rolle's Theorem guarantees the existence of some c in (0, 1) with f ' (c) = 0? If not, explain why not. |
Exercise 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).
Exercise 2
For the function f(x) = 2x, 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.
Exercise 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) = x2 for -1 < x < 1 and f(x) = 3 for x ≥ 1)
Exercise 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)
Exercise 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.)
Exercise 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) = x3 on the interval (-2, 2)
Exercise 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)
Exercise 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)
Exercise 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).