Verify Rolles theorem for function `f(x)=log(x^2+2)-log3` on `[-1,\ 1]`

Answer

N/a

Explanation

We have, ` f(x) = log(x^(2)+2)-log3`.
(i) Logarithmic functions are continous in their domain.
Hence, `f(x) = log(x^(2)+2).2x-0`
`= (2x)/(x^(2)+2)`, which exists in `(-1,1)`
Hence, `f(x)` is differentiable in `(-1,1)`.
(iii) `f(-1) = log[(-1)^(2)+2]-log3=log3-log3=0` and
`f(1)=log(1^(2)+2)-log3=log3-log3=0`
`rArr f(-1)=f(1)`
Conditions of Rolle's theorem are satisfied.
Hence, there exists a real number c such that
`f(c) = 0`
` rArr (2c)/(c^(2)+2) = 0`
`rArr c = 0 in (-1,1)`
Hence, Rolle's theorem has been verified.