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Derivatives

Derivatives

At a Glance - Rolle's Theorem

Rolle's Theorem says:

Let f be a function that

  • is continuous on the closed interval [ab]
      
  • is differentiable on the open interval (ab), and
      
  • has (a) = (b).

Then there is some c in the open interval (ab) with f ' (c) = 0.

Sometimes the third condition is stated as (a) = (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, ' (c) is 0 for every value of c in the interval (ab).

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 (ab). 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 [ab]. Then there doesn't need to be any c in (ab) with ' (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 ' (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 ' (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) = x2x. 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).

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