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# Calculus

Applications of the Derivative

# Rolle’s Theorem

Page 1
Problems 1-2
Page 2
Problems 3-7

Rolle’s theorem states that any real differentiable function that attains equal values at the endpoints of an interval must have in this interval at least one stationary point, that is a point, at which the first derivative is zero. Geometrically, this means that the tangent to the graph of the function at this point is horizontal (Figure $$1$$).

This property was known in the $$12$$th century in ancient India. The outstanding Indian astronomer and mathematician Bhaskara $$II$$ $$\left(1114-1185\right)$$ mentioned it in his writings.

Figure 1.

Figure 2.

In a strict form this theorem was proved in $$1691$$ by the French mathematician Michel Rolle $$\left(1652-1719\right)$$ (Figure $$2$$).

In modern mathematics, the proof of Rolle’s theorem is based on two other theorems − the Weierstrass extreme value theorem and Fermat’s theorem. They are formulated as follows:

### The Weierstrass extreme value theorem

If a function $$f\left( x \right)$$ is continuous on a closed interval $$\left[ {a,b} \right],$$ then it attains the least upper and greatest lower bounds on this interval.

### Fermat’s theorem

Let a function $$f\left( x \right)$$ be defined in a neighborhood of the point $${x_0}$$ and differentiable at this point. Then, if the function $$f\left( x \right)$$ has a local extremum at $${x_0},$$ then

$f’\left( {{x_0}} \right) = 0.$

Consider now Rolle’s theorem in a more rigorous presentation. Let a function $$y = f\left( x \right)$$ be continuous on a closed interval $$\left[ {a,b} \right],$$ differentiable on the open interval $$\left( {a,b} \right),$$ and takes the same values at the ends of the segment:

$f\left( a \right) = f\left( b \right).$

Then on the interval $$\left( {a,b} \right)$$ there exists at least one point $$\xi \in \left( {a,b} \right),$$ in which the derivative of the function $$f\left( x \right)$$ is zero:

$f’\left( \xi \right) = 0.$

#### Proof.

If the function $$f\left( x \right)$$ is constant on the interval $$\left[ {a,b} \right],$$ then the derivative is zero at any point of the interval $$\left( {a,b} \right),$$ i.e. in this case the statement is true.

If the function $$f\left( x \right)$$ is not constant on the interval $$\left[ {a,b} \right],$$ then by the Weierstrass theorem, it reaches its greatest or least value at some point $$\xi$$ of the interval $$\left( {a,b} \right),$$ i.e. there exists a local extremum at the point $$\xi.$$ Then by Fermat’s theorem, the derivative at this point is equal to zero:

$f’\left( \xi \right) = 0.$

Rolle’s theorem has a clear physical meaning. Suppose that a body moves along a straight line, and after a certain period of time returns to the starting point. Then, in this period of time there is a moment, in which the instantaneous velocity of the body is equal to zero.

## Solved Problems

Click on problem description to see solution.

### ✓Example 1

Prove that if the equation

$f{\left( x \right) = {a_0}{x^n} + {a_1}{x^{n – 1}} + \ldots }\kern0pt{\;+\;{a_{n – 1}}x = 0\;\;\;}$

has a positive root $$x = {x_0},$$ then the equation

$n{a_0}{x^{n – 1}} + \left( {n – 1} \right){a_1}{x^{n – 2}} + \ldots + {a_{n – 1}} = 0$

also has a positive root $$x = \xi,$$ where $$\xi \lt {x_0}.$$

### ✓Example 2

Check the validity of Rolle’s theorem for the function

$f\left( x \right) = {x^2} – 6x + 8.$

### ✓Example 3

Check the validity of Rolle’s theorem for the function

$f\left( x \right) = \sqrt {1 – {x^2}}$

on the segment $$\left[ { – 1,1} \right].$$

### ✓Example 4

Check the validity of the Rolle’s theorem for the function

$f\left( x \right) = \frac{{{x^2} – 4x + 3}}{{x – 2}}$

on the segment $$\left[ {1,3} \right].$$

### ✓Example 5

Check the validity of Rolle’s theorem for the function

$f\left( x \right) = \left| {x – 1} \right|$

on the segment $$\left[ {0,2} \right].$$

### ✓Example 6

Determine the number of stationary points of the function

$f\left( x \right) = x\left( {x – 1} \right)\left( {x – 2} \right)$

and indicate the intervals, in which they are located.

### ✓Example 7

Check the validity of Rolle’s theorem for the function

$f(x) = \begin{cases} x^2, & \text{if}\;\;\; 0 \le x \le 2 \\ 6-x, & \text{if}\;\;\; 2 \lt x \le 6 \end{cases}.$

### Example 1.

Prove that if the equation

$f{\left( x \right) = {a_0}{x^n} + {a_1}{x^{n – 1}} + \ldots }\kern0pt{\;+\;{a_{n – 1}}x = 0\;\;\;}$

has a positive root $$x = {x_0},$$ then the equation

$n{a_0}{x^{n – 1}} + \left( {n – 1} \right){a_1}{x^{n – 2}} + \ldots + {a_{n – 1}} = 0$

also has a positive root $$x = \xi,$$ where $$\xi \lt {x_0}.$$

#### Solution.

In addition to $$x = {x_0},$$ the first equation has the root $$x = 0.$$ Consequently, the function $$f\left( x \right)$$ satisfies the conditions of Rolle’s theorem:

$f\left( 0 \right) = f\left( {{x_0}} \right) = 0.$

The second equation is obtained by differentiating the first equation:

${f’\left( x \right) } = {{\left( {{a_0}{x^n} + {a_1}{x^{n – 1}} + \ldots + {a_{n – 1}}x} \right)^\prime } } = {n{a_0}{x^{n – 1}} + \left( {n – 1} \right){a_1}{x^{n – 2}} + \ldots + {a_{n – 1}} = 0.}$

According to Rolle’s theorem, there is an interior point $$x = \xi$$ on the interval $$\left[ {0,{x_0}} \right]$$ where the derivative is zero. Consequently, the point $$x = \xi$$ is a solution of the second equation where $$0 \lt \xi \lt {x_0}.$$

### Example 2.

Check the validity of Rolle’s theorem for the function

$f\left( x \right) = {x^2} – 6x + 8.$

#### Solution.

The given quadratic function has roots $$x = 2$$ and $$x = 4,$$ that is

$f\left( 2 \right) = f\left( {4} \right) = 0.$

The by Rolle’s theorem, there is a point $$\xi$$ in the interval $$\left( {2,4} \right)$$ where the derivative of the function $$f\left( x \right)$$ equals zero.

Calculate the derivative:

$f’\left( x \right) = {\left( {{x^2} – 6x + 8} \right)^\prime } = 2x – 6.$

It is equal to zero at the following point $$x = \xi:$$

${f’\left( x \right) = 0,\;\;}\Rightarrow { 2x – 6 = 0,\;\;}\Rightarrow {x = \xi = 3.}$

It can be seen that the resulting stationary point $$\xi = 3$$ belongs to the interval $$\left( {2,4} \right)$$ (Figure $$3$$).

Figure 3.

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Problems 1-2
Page 2
Problems 3-7