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Table of contents
1.
Introduction
2.
Rolle's Theorem
3.
Graphical representation of Rolle's Theorem
4.
Proof of Rolle's Theorem
5.
Example of Rolle's Theorem
6.
FAQs
7.
Key Takeaways
Last Updated: Mar 27, 2024

Rolle's mean value theorem

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Introduction

Rolle's mean value theorem is a special case of Lagrange's mean value theorem. It states that if a function is continuous on the closed interval [a,b] and differentiable in the open interval (a,b), such that f(a)=f(b) where 'a' and 'b' are some real numbers, then there exists a point c in the range (a,b) such that [f' (c)=0]. Now, let's understand this statement thoroughly.

Rolle's Theorem

If a function f(x) is defined in the interval (a,b) meaning from x=a to x=b, and the function satisfies the given below criteria then there is a point between x=a to x=b such that f’(c)=0. f’(c)=0 means that at x=c when we draw a tangent to the curve f(x) then the tangent is a horizontal line parallel to the x-axis. The conditions which are needed to be satisfied by f(x) are given below:

  1. The function f(x) should be continuous in the closed interval [a,b]
  2. f(x) should be differentiable in the open interval (a,b)
  3. f(a)=f(b) 

The main objective of Rolle's theorem is to prove that there will always be a point x=c from x=a to x=b such that the tangent on that point will be parallel to the x-axis.

To understand Rolle's theorem better, we will discuss the graphical representation of the above statement.

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Graphical representation of Rolle's Theorem

Source

In the graph given above, we can see that the function is continuous in the range [a,b] at every point. And we can draw a tangent to the curve at every point between x=a to x=b. We can also observe that the 'y' coordinate for x=a and x=b is the same. All three conditions of Rolle's theorem are satisfied, so according to this theorem, a point must exist where a tangent is drawn, then the tangent will be parallel to the x-axis. We can verify this from the graph. At point x=c, we can observe the tangent at that point is parallel to the x-axis.

Proof of Rolle's Theorem

We have to prove that if a function f(x) is continuous in [a,b] and differentiable in (a,b) then there always exists a point 'c' such that f' (c)=0.

Let's consider the possible cases that we can encounter for a curve satisfying all the three conditions of Rolle's theorem.

  • Case-1

fig(1)

f(x) is the same for all values in the range [a,b]. Here we can pick any point from 'a' to 'b,' and if we draw a tangent, it will always be parallel to the x-axis, as seen in fig(1). This is because the curve we are discussing is constant. The value of f(x) is constant for all the values of x.

f(x) = K

And we know that the differentiation of a constant is 0.

f’(x)=0

Therefore at every point between 'a' and 'b,' we can take a 'c' where f' (c)=0

  • Case-2

f(x) has a different value than f(a) at some x between 'a' and 'b'. The extreme value theorem states that the function f attains a maximum or minimum value between 'a' and 'b.' 

fig(2)

When f(x) is increasing after x=a then it has to start decreasing before x=b so that it can again reach the level of f(a) (because f(a)=f(b)) as seen in fig(2). The point where the curve starts decreasing is the local maxima of the curve; at that point, f' (x)=0.

fig(3)

In the same way, when f(x) starts decreasing after x=a, it has to start increasing before x=b to reach the level of f(a) again, as seen in fig(3). The point where the curve begins rising will be the local minima; at that point, f' (x)=0.

Now, these two cases discussed above cover all the possible scenarios, so it can be stated that we always have a point 'c' where f' (c)=0 when the curve f(x) satisfies all the three conditions of Rolle's theorem. 

Note: A curve must satisfy all the three conditions of Rolle's theorem to apply the above proof.

Let's discuss some examples of Rolle's theorem to give you an insight on how to solve the questions on Rolle's theorem.

Example of Rolle's Theorem

Example: Verify Rolle’s theorem for the function f(x) = x2 - 4, x∈ [-2,+2]

Solution: First we have to check the below-given conditions to verify Rolle’s theorem for f(x).

  1. f(x) should be continuous in [a,b]
  2. f(x) should be differentiable in (a,b)
  3. f(a)=f(b)
  • Condition 1:

The given function is polynomial, and all polynomial functions are continuous for all x∈R. So f(x) satisfies the first condition of Rolle's theorem.

  • Condition 2:

Polynomial functions are also differentiable for all x∈R. Therefore, f(x) satisfies the second condition for Rolle's theorem.

  • Condition 3:

We have to check if f(a)=f(b). Here a=-2 and b=2.

f(a)=f(-2)=0

f(b)=f(2)=0

Hence f(a)=f(b)

So the third condition of Rolle's theorem is also satisfied by f(x).

According to Rolle's theorem, if a function satisfies all three conditions, there is a point 'c' such that f' (c)=0.

Now,

f’(x)=d(f(x))/dx

f’(x)=2x

If f’(x)=0 then 2x=0

This implies that x=0

Therefore c=0

And -2<0<2

Or a<c<b

So there is a point 'c' between x=a and x=b such that f' (c) is zero.

Hence Rolle's theorem is verified for given f(x).

FAQs

  1. What are the three conditions for Rolle's theorem?
    The three conditions for Rolle's theorem are listed below.
    1. The function f(x) should be continuous in the closed interval [a,b]
    2. f(x) should be differentiable in the open interval (a,b)
    3. f(a)=f(b) 
  2. What does Rolle's theorem state?
    Rolle’s theorem state that if a function is continuous in [a,b], differentiable in (a,b) and f(a)=f(b) then there exists a ‘c’ where f’(c)=0.
  3. What is Lagrange's mean value theorem?
    Lagrange's mean value theorem states that if f(x) is continuous in [a,b] and differetiable in (a,b) then there exists a ‘c’ such that f’(c) = [f(b)-f(a)]/(b-a).
  4. How are Lagrange's mean value theorem and Rolle's theorem related?
    Rolle's theorem is a particular case of Lagrange's mean value theorem.
  5. What is the first mean value theorem?
    Lagrange's mean value theorem is the first mean value theorem.

Key Takeaways

In this article, we have extensively discussed Rolle’s mean value theorem. We hope that this blog has helped you enhance your engineering mathematics and statistics knowledge. Check out our articles on the following topics related to engineering mathematics if you want to learn more.

  1. Mean, Variance, Standard Deviation
  2. Cauchy’s Mean Value Theorem
  3. Poisson Distribution
  4. Prosecutor’s fallacy
  5. Initial Value Theorem

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