Question

Can you answer this question

In the function ?(?) = ? − 2ln (?) over the interval [1, ?] is Rolle’s Theorem applied or not and explain why?

Answer 1

For Rolle's theorem to be applicable to a function on an interval [a,b] ,the following three conditions should be met :

1.) it should be continuous on the closed interval [a,b],

2) it should be differentiable on the open interval (a,b) .

3) the value of two functions i.e, f(a) should be equal to f(b) .

Now in the given function f(x)=x-2lnx , this is the sum of a polynomial and logarithmic function .The logarithmic functions are differentiable everywhere on their domain and so are the polynomial functions .We know that sum of two continuous functions is also continuous and also differentiable .

Hence the given function is differentiable on (1,e) and continuous on [1,e] .

Now we need to check the third condition ,

f(x)=x-2lnx

f(1)= 1-2ln(1) =1-0 =1

f(e) =e-2lne =e-2

Thus , f(1) not equal to f(e) .thus the third condition is not satisfied .

Hehce rolle's theorem is not applicable to the given function .