Cauchy Mean Value Theorem

The Mean Value Theorem of Cauchy is a generalisation of Lagrange’s Mean Value Theorem.

The Extended or Second Mean Value Theorem is another name for this Theorem. It provides a

connection between two functions’ derivatives and changes in these functions over a specified

interval. If a function f (x) is continuous in the close interval [a, b] where (a≤x ≤b) and differentiable in the open interval [a, b] where (a < x< b), then there must be at least one point x = c on this interval, as described by the normal mean value theorem given as :

f(b) – f (a) = f’ (c) (b-a). This article will prove Cauchy’s Mean Value Theorem and look at a few examples.

How is Cauchy Mean Theorem different from Lagrange Mean Theorem and Rolle’s Theorem?

Rolle’s Theorem is a notable example of the mean value theorem, satisfying certain requirements. Lagrange’s mean value theorem is simultaneously the mean value theorem and the first mean value theorem. The mean can be defined as the average of a set of values. The procedure of determining the mean value of two separate functions is different in the case of integrals. Let’s look at Rolle’s Theorem and the Mean Value and geometrical meaning of such functions.

State and Proof Cauchy’s Mean Value Theorem

The state and prove Cauchy’s mean value theorem analysis:

If a function f(x) and g(x) be continuous on an interval [a,b] , differentiable on (a,b), and g'(x) is

not equal to 0 for all x ε (a,b). Then there is a point x = c in this interval given as :

f(b)- f(a)   =  f'(c)

g(b)- g(a)     g'(c)

Proof

The left side of the Cauchy formula’s denominator is not zero in this case: g(b)-g(a) is not equals

to 0. If g(b) = g(a), Rolle’s theorem states that there is a point d? (a,b) where g'(d) = 0. As a

result , contradicts the hypothesis that g'(x) ≠ 0 for all x ? (a,b).

Let’s now use the auxiliary function.

F (x) = f (x) + λg(x)

And select λ in such a way to satisfy the given condition

F (a) = f (b). we get,

f (a) + λg(a) = f (b) + λg(b)

= f(b)-f (a) = λ[g(a)- g(b)]

f(b)- f(a) =   f'(c)

g(b)- g(a)      g'(c)

And the function F (x) exists in the form

F(x)  = f(x) – f(b) – f(a) g(x)

               g(b) – g(a)

F (x) is continuous in the closed interval (a≤x ≤b), differentiable in the open interval (a < x< b),

and takes equal values at the interval’s endpoints. As a result, it fulfils all of Rolle’s Theorem

requirements. Then, in the interval (a,b), a point c exists.

F’ (c) is equal to zero.

So ,

f'(c) – f(b) – f(a) g'(c) = 0

           b- a

Or

f(b) – f(a) = f'(c)

g(b) – g(a)  g'(c)

Now putting g(x) = x ,

there comes the Lagrange formula :

f'(c) – f(b) – f(a)

             b-a

The geometric meaning of Cauchy’s mean value theorem is as follows.

What if the parametric equations result in a curve? X = f (t) and Y = g (t), 

where t is a positive integer in the range [a,b]. 

The point of the curve in the following figure runs from A (f) to B (f) as the parameter t is changed (a). B (f(b), g (b) to g(a) 

Is there a point (f(c), g(c)) on the curve, according to Cauchy’s mean value theorem? The tangent is perpendicular to the chord connecting the curve’s ends, A and B.

Cauchy Mean Value Theorem Examples

Ques 1. Calculate the value of x, which satisfies the Mean Value Theorem for the following function

F(x) = x2 + 2x + 2

Explanation:

Given

f(x) = x2 + 2x + 2

According to Mean Value theorem,

2c = -7

C = -7/2

Ques 2. Calculate the value of x which satisfies the Mean Value Theorem for the following function

F(x) = x2 + 4x + 7

Explanation:

Given

f(x) = x2 + 4x + 7

According to Mean Value theorem,

Conclusion

The Mean Value Theorem of Cauchy is a generalisation of Lagrange’s Mean Value Theorem. The Extended or Second Mean Value Theorem is another name for this Theorem. So Far in this article, we have discussed the statement of the Cauchy Theorem, Proved it and solved Cauchy mean value theorem examples.I hope this article helped clarify the confusion between the theorems, and wherever two functions appear, we can use this Theorem to solve and find x.