In: Advanced Math
State and prove the Generalised Mean Value Theorem.
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.