Question

In: Advanced Math

State and prove the Generalised Mean Value Theorem.

State and prove the Generalised Mean Value Theorem.

Solutions

Expert Solution

Generalized Mean Value Theorem:

Let f(x) is a function such that,

• It is continuous on [a,b].

• It is differentiable on (a,b).

then there c (a,b) such that

This theorem can be generalized by using Cauchy mean value theorem which can also be called generalized mean value theorem.

Theorem Statement:

Let f(x) and g(x) be the functions defined such that,

• They are continuous on [a,b]

• They are differentiable on (a,b)

and then there c (a,b) and , such that

Proof:

As per the theorem statement,

f(x) and g(x) are functions defined in such a way that,

• Continuous on [a,b]

• Differentiable in (a,b)

Let us assume h(x) is a function that holds same qualities and where S is some constant such that, is holds good.

Since, h(x) is continuous on [a,b] and differentiable in (a,b) and h(a)=h(b) then it obeys Rolle's theorem.

From (i) and (ii) we can conclude that,

Hence proved.


Related Solutions

State and prove the Weighted Mean Value Theorem for integrals.
State and prove the Weighted Mean Value Theorem for integrals.
Use the Intermediate Value Theorem and the Mean Value Theorem to prove that the equation cos...
Use the Intermediate Value Theorem and the Mean Value Theorem to prove that the equation cos (x) = -10x has exactly one real root. Not permitted to use words like "Nope", "Why?", or "aerkewmwrt". Will be glad if you can help me with this question, will like to add some of your points to the one I have already summed up.. Thanks
state and prove spectral mapping theorem
state and prove spectral mapping theorem
State and prove spectral mapping theorem
State and prove spectral mapping theorem
State and prove spectral mapping theorem
State and prove spectral mapping theorem
state and prove Theorem Holomorphy of Rλ
state and prove Theorem Holomorphy of Rλ
State and prove Lebesgue Dominated Convergence theorem.
State and prove Lebesgue Dominated Convergence theorem.
Use the Mean-Value Theorem to prove that 30/203 < √ 103 − 10 < 3 /20...
Use the Mean-Value Theorem to prove that 30/203 < √ 103 − 10 < 3 /20 .
Explain what it is a neutral theorem in Euclidean geometry. State & prove both: the theorem...
Explain what it is a neutral theorem in Euclidean geometry. State & prove both: the theorem on construction of parallel lines and its converse. Which one of them is neutral?
1. State the prove The Density Theorem for Rational Numbers.
  Question 1. State the prove The Density Theorem for Rational Numbers. Question 2. Prove that irrational numbers are dense in the set of real numbers. Question 3. Prove that rational numbers are countable Question 4. Prove that real numbers are uncountable Question 5. Prove that square root of 2 is irrational
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT