# CBSE Class 12-science Answered

**Verify Rolle's theorem for the function f (x) = on [a, b], where 0 < a < b.**

Asked by Topperlearning User | 04 Aug, 2014, 03:48: PM

Expert Answer

We have,

*f*(

*x*) = = log (

*x*

^{2}+

*ab*) - log

*x*- log (

*a + b*)

Since logarithimic function is differentiable and so continuous on its domain. Therefore,

*f*(*x*) is continuous on [*a, b*] and differentiable on (*a, b*).Also,

*f*(*a*) =_{}= log 1 = 0, and*f*(*b*) =_{}= log 1 = 0\

*f*(*a*) =*f*(*b*)Thus, all the three conditions of Rolle's theorem are satisfied.

Now, we have to show that there exists

*c*ε (*a, b*) such that*f*'(*c*) = 0We have,

*f*(

*x*) = log (

*x*

^{2}+

*ab*) - log

*x*- log (

*a + b*)

*f*'(

*x*) =

\

*f '*(*x*) = 0 & = 0 =>*x*^{2}=*ab*=>*x*=Since a<<b Therefore,

*c*= ε (*a, b*) is such that*f*'(*c*) = 0Hence, Rolle's theorem is verified.

Answered by | 04 Aug, 2014, 05:48: PM

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