## 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) = x**^{2}+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) = 3x**^{2} 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.

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