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Table of contents
1.
Introduction
2.
Normal Mean Value Theorem
3.
Cauchy's Mean Value Theorem
3.1.
Proof of Cauchy Mean Value Theorem
3.2.
Derivation of Lagrange Mean Value Theorem from Cauchy's Mean Value Theorem
3.3.
Examples of the Mean Value Theorem
4.
FAQs
5.
Key Takeaways
Last Updated: Mar 27, 2024
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Cauchy's Mean Value Theorem

Author HET FADIA
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Introduction

In this article, we will learn Cauchy's Mean Value theorem. The theorem is also known as the Second Mean Value theorem. It tells us relationships between derivatives of two functions on a finite interval.

Before we see Cauchy's Mean Value Theorem, let us look at the Normal Mean Value Theorem.

Normal Mean Value Theorem

The theorem states that if a function f(x) is continuous in a closed finite interval [a,b] where a<=x<=b and differentiable in the open interval (a,b) where a<=x<=b then there exists at least one point c where a<=c<=b such that f'(c) x (b-a) = f(b) - f(a).

In other words, we can say that the differentiation at a point c where a<=c<=b is equal to the slope of the line joining a to b.

The Normal Mean Value Theorem is also known as the Lagrange Mean Value Theorem.

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Cauchy's Mean Value Theorem

The theorem states the relationship between the derivative of two functions on a finite interval.

For two functions f(x) and g(x), the theorem states that if both the functions are differentiable in the closed finite interval [a,b] where a<=x<=b and differentiable in the open interval (a,b) where a<=x<=b and g'(x) is not equal to 0 where x belongs to the interval (a,b) then there exists at least one point c where a<=c<=b such that 

f'(c) / g'(c)=(f(b)-f(a))/(g(b)-g(a))

Proof of Cauchy Mean Value Theorem

First g(b)-g(a) can not be zero. Because if g(b)-g(a)=0 then by Roll's theorem there exists a point x such that g'(x)=0 where a<x<b. And it is given that g'(x) is not equal to 0 in the interval a<=x<=b. Now we take a function F(x) such that

F(x) = f(x) + k g(x) 

Here we have chosen k such that F(a)=F(b).

Thus substituting we get

f(a) + k g(a) = f(b) + k g(b)

Thus f(a) - f(b) = k x (g(b) - g(a))

Thus we get the value k= (f(a)- f(b))/(g(b)-g(a))

Now substituting the value of k in F(x) = f(x) + k g(x) we get,

F(x) = f(x) + (f(a)- f(b))/(g(b)-g(a))x g(x)

Thus F(x) = f(x) - (f(b)- f(a))/(g(b)-g(a))x g(x)

F(a) = F(b) and F(x) is continuous in the closed interval [a,b] and differentiable in the open interval (a,b). Thus by Rolle's theorem, we get F'(c) = 0.

Hence 

f'(c) - (f(b)- f(a))/(g(b)-g(a)) x g'(x)=0

Simplifing the above equation we get

(f(b)-f(a))/(g(b)-g(a))= f'(c)/g'(c)

Thus we proved Cauchy's Mean Value Theorem.

Derivation of Lagrange Mean Value Theorem from Cauchy's Mean Value Theorem

By substituting g(x)=x in Cauchy's Mean Value Theorem, we get Lagrange Mean Value Theorem.

Lagrange Mean Value theorem states that for continuous function f(x) and g(x) on the interval [a,b] and differentiable in the open interval (a,b), there exists a point c such that 

(f(b)-f(a))/(b-a)= f'(c)

Examples of the Mean Value Theorem

1. Find the value of c which satisfies the Mean Value Theorem for the  function f(x) = x2+2x+1 in the interval [1,2].

Using Cauchy's Mean Value Theorem we get

(f(b)-f(a))/(g(b)-g(a))= f'(c)/g'(c)

Substituting g(x)=x in the above theorem, we get

 f'(c)=(f(b)-f(a))/(b-a)

Now f'(x)=2x+2. Thus f'(c)=2c+2

Also, f(b)=4+4+1= 9  and f(a)=1+2+1=4.

Thus (f(b)-f(a))/(b-a)=(9-5)/(2-1)=4

Equating both the sides we get 2c+2=4 => c=1

Ans: c=1

2. What is the value of c for the function f(x) = 4x and g(x) = 3x2 which lies in the interval [1,2]?

Using Cauchy's Mean Value Theorem we get

(f(b)-f(a))/(g(b)-g(a))= f'(c)/g'(c)

f'(c)=4 and g'(c)=6c

f(b)=4 x 2 = 8

f(a)=4 x 1 = 4

g(b)=3 x 4= 12

g(a)=3 x1=3

Thus substituting the above values in the theorem, we get

(8-4)/(12-3)=4/6c

Thus c=3/2

FAQs

1. What is Cauchy's Mean Value Theorem?

For two functions f(x) and g(x), the theorem states that if both the functions are differentiable in the closed finite interval [a,b] where a<=x<=b and differentiable in the open interval (a,b) where a<=x<=b and g'(x) is not equal to 0 where x belongs to the interval (a,b) then there exists at least one point c where a<=c<=b such that 

(f(b)-f(a)) / (g(b)-g(a))=f'(c) / g'(c)

2. What are the necessary conditions for Cauchy's Mean Value Theorem to be applicable on the functions f(x) and g(x) in the interval [a,b]?

For Cauchy's Mean Value Theorem, the two functions f(x) and g(x) must be continuous in the interval [a,b] and must be differentiable in the open interval (a,b).

3. How can we derive Lagrange Mean Value Theorem using Cauchy's Mean Value Theorem?

In Cauchy' mean Value Theorem

(f(b)-f(a)) / (g(b)-g(a))=f'(c) / g'(c)

Suppose we take g(x)=x, then g(b)=b, g(a)=a and g'(x)=1.

Substituting g(x)=x, we get Lagrange Mean Value Theorem

(f(b)-f(a)) / (b-a)=f'(c) 

4. What relationship does Cauchy's Mean Value Theorem give us about the functions?

Cauchy's Mean Value Theorem tells us relationships between derivatives of two functions on a finite interval.

5. What are the different names of Cauchy's Mean Value Theorem?

Cauchy's Mean Value Theorem is also known as the Second Mean Value Theorem or the Extended Mean Value Theorem.

Key Takeaways

In this article, we have extensively discussed Cauchy's Mean Value Theorem along with its proof. We also derived the Lagrange Mean Value theorem using Cauchy's Mean Value Theorem.

We hope that this blog has helped you enhance your knowledge regarding Mean Value Theorem, and if you would like to learn more, check out our articles on

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