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).
Answer
We need to check that f satisfies all the hypotheses of Rolle's Theorem.
f(x) = sin(x) is continuous everywhere, so f is continuous on [0, 2π].
f is differentiable everywhere, therefore it's differentiable on (0, 2π).
f(0) = 0 and f(2π) = 0, so f(0) = f(2π).
Since f satisfies all the hypotheses of Rolle's Theorem, there is some value of c in (0, 2π) with f ' (c) = 0.By looking at the graph of f(x) = sin(x), we see that there are two such values of c:
If any one of the hypotheses of Rolle's Theorem fails, the conclusion can fail too.
Example 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.
Answer
No. f is continuous on [a, b], and differentiable on (a,b) (we don't know the derivative just yet, but we have a nice, smooth curve so the function is differentiable). The third hypothesis is false, though. We don't have f(a) = f(b). Rolle's Theorem can't guarantee that there is some c in (a, b) with f ' (c) = 0.
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) = x2 for -1 < x < 1 and f(x) = 3 for x ≥ 1)
Answer
Yes. While f is not continuous overall, f is continuous on [-1, 1]. f is differentiable on (-1, 1) and f(a) = f(b), Rolle's Theorem says there is some c in (-1, 1) with f ' (c) = 0.
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)
Answer
Yes. f is not differentiable on [a, b], but it is differentiable on (a, b) so we can totally use Rolle's Theorem here.We don't need the function to be differentiable at the endpoints of the interval, just at all the numbers between the endpoints.
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.)
Answer
No. f(a) ≠ f(b), so we aren't allowed to use Rolle's Theorem. It so happens that there is a point in the interval where the value of the derivative is 0 anyway, but we didn't use Rolle's Theorem to figure that out.
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) = x3on the interval (-2, 2)
Answer
No. f(-2) ≠ f(2), so we can't use Rolle's Theorem.
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)
Answer
Yes. f(x) = cos(x) is continuous and differentiable, and cos(-π) = 1 = cos(3π), so Rolle's Theorem guarantees the existence of c in (-π, 3π) with f ' (c) = 0.
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)
Answer
No. f(-2) = 20, but f(2) = 4. Since f(-1) ≠ f(2), we can't use Rolle's Theorem.
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).
Answer
No. f(x) is not continuous on [-1, 1] because f(-1) is undefined. We can't use Rolle's Theorem.
on the interval (-1, 1).