Cauchy’s Mean Value Theorem and L’Hospital’s Rule

Cauchy’s mean value theorem is a generalisation of the conventional mean value theorem, which is a well-known mathematical result. This theorem is often referred to as the Extended Mean Value Theorem or the Second Mean Value Theorem. It is defined as follows: If a function (x) is continuous in the close interval [a, b] where (x b) and differentiable in the open interval [a, b] where (x b), then there is at least one point on this interval where x = c, which is denoted by the symbol c.

f(b) – f (a) =f’ (c) (b-a)

A finite interval is established by establishing the relationship between the derivatives of two functions and the changes in these functions over time.

Consider the functions f(x) and g(x) to be continuous on the interval [a,b], differentiable on the interval (a,b), and g'(x) to be not equal to 0 for all x ε(a,b). Then there is a point x = c in the interval given as

f(b)-f(a)              f'(c)

_______    =     _____

g(b)-g(a)           g'(c)    

Proof of Cauchy’s mean value theorem

f(b)-f(a)             f'(c)

_______   =    _____

g(b)-g(a)           g'(c)    

Since g(b)-g(a) is not equal to zero, the denominator in the left side of the Cauchy formula is not zero. If g(b) equals g(a), then Rolle’s theorem states that there is a point d? (a,b) at which g'(d) equals zero. As a result, does the premise that g'(x) = 0 for all x contradict itself? (a,b).

We are now going to 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)]

λ = f(b)-f (a)/g(a)- g(b)

And the function F (x) exists in the form

                        f(b)-f (a)

 F (x)=  f(x) –   _______.   g(x)

                       g(b)- g(a)

It is true that the function F (x) is continuous in the closed interval [a , b], differentiable in the open interval (a , b), and that it takes equal values at both ends of the interval. As a result, it satisfies all of the requirements of Rolle’s theorem. Afterwards, in the interval (a,b) defined as follows, there exists a point c

F’ (c ) =0

Therefore it gives that

             f(b)-f (a)

 f'(c) –  _______.   g'(c)

               b- a

Or 

f(b)-f(a)           f'(c)

_______   =  _____

g(b)-g(a)         g'(c)    

The Lagrange formula is obtained by substituting g (x) = x into the preceding formula:

             f(b)-f (a)

 f'(c) –  _______.  g'(c)

               b- a

Cauchy’s mean value theorem has the geometric meaning that has been provided. Consider the fact that the parametric equations produce a curve. X = f (t) and Y = g (t), where the parameter t is in the range [a,b], and t is the time.

Changing the value of the parameter t causes the point of the curve in the given figure to move away from A (f) (a). g(a) to B (f(b), g (b)) is a transformation.

There is a point (f(c) and g(c)) on the curve, according to Cauchy’s mean value theorem? where the tangent is parallel to the chord connecting the two endpoints of the curve, A and B, and

L’Hospital’s rule 

L’Hospital’s rule is a general way of evaluating indeterminate forms such as 0/0 or ∞/∞ that can be applied to any situation. L’Hospital’s rule is used in calculus to determine the limits of indeterminate forms for the derivatives of a function. The L-Hospital rule can be applied more than once in a single situation. You can continue to apply this rule as long as it retains any indefinite form after its initial application. It is impossible to apply L’Hospital’s Rule to an issue that does not fall into one of the indeterminate forms.

The L’Hospital’s Rule proof

L’Hospital’s rule can be demonstrated through the use of the Extended Mean Value Theorem or Cauchy’s Mean Value Theorem.

For two continuous functions on the interval [a, b] that are also differentiable on the interval (a, b), the result is the following:

f'(c )/g'(c )= [f(b)-f(a)]/[g(b)-g(a)] such that c is a member of (a, b).

Pretend the two functions f and g are defined on the interval (c, b) in such a way that f(x)→0 and g(x)→0 , as x→c+.

However, we have the fact that f'(c) / g'(c) tends to finite limits. In this case, the functions f and g are differentiated, and the functions f'(x) and g'(x) exist on the set [c, c+k], respectively. Additionally, the functions f’ and g’ are continuous on the interval [c, c+k], provided that the conditions f(c), g(c), and g'(c) are met.

Using the Cauchy Mean Value According to the theorem, there exists ck∈ (c, c+k), which is such that

In mathematics, 

f'(ck)/g'(ck)= [f(c+k)-f(c)]/[g(c+k)-g(c)] = f(c+k) and g(c+k)

Now, using k→0+,

limk→0+= f'(ck)/g'(ck) =limx→c+ f'(x)/g'(x)

So, we get 

limx→c+= f(x)/g(x) =limx→c+ f'(x)/g'(x)

Conclusion

Cauchy’s mean value theorem is a generalisation of the conventional mean value theorem, which is a well-known mathematical result. This theorem is often referred to as the Extended Mean Value Theorem or the Second Mean Value Theorem. It is defined as follows: If a function (x) is continuous in the close interval [a, b] where (x b) and differentiable in the open interval [a, b] where (x b), then there is at least one point on this interval where x = c.L’Hospital’s rule is a general way of evaluating indeterminate forms such as 0/0 or ∞/∞ that can be applied to any situation. L’Hospital’s rule is used in calculus to determine the limits of indeterminate forms for the derivatives of a function. The L-Hospital rule can be applied more than once in a single situation. You can continue to apply this rule as long as it retains any indefinite form after its initial application.