The function in which Rolle’s theorem is verified is:

Options

A. `f(x)=log.((x^(2)+ab))/((a+b)x)" in "[a, b] ("where "0ltaltb)`
B. `f(x)=(x-1)(2x-3)" in "[1, 3]`
C. `f(x)=2+(x-1)^(2//3)" in "[0,2]`
D. `f(x)=cos.(1)/(x)" in [-1, 1]`

Answer

A

Explanation

`f(x)=log(x^(2)+ab)-log(a+b)-logx`
f(x) is continuous in [a, b]
Now, `f'(x)=[(2x)/(x^(2)+ab)-(1)/(x)]`. which clearly exists in `(a, b) and f (a) = f (b) = 0`
So Rolle's theorem is verified.
(B) `f(1)ne f(3)`. So Rolle's theorem not satisfied
(C) `f'(x)=(2)/(3(x-1)^(1//2)).` Clearly `f'(1)` Clearly does not exist, so Rolle's theorem is not satisfied.
(D) `f(x)=cos.(1)/(x)` is discontinuous at x = 0, so Rolle's theorem is not satisfied