Verify Rolle's theorem for the function f(x) = {log `(x^(2) +2) - log 3`} in the interval `[-1,1]`.

Explanation

f(x) = {log `(x^(2) + 2) - log 3`}
(i) f(x) has a definite and unique value at each point in `[-1,1]`.
`therefore` It is continuous in `[-1,1]`.
(ii) `f'(x) = (2x)/(x^(2) + 2) - 0 = (2x)/(x^(2) + 2)`
Which is defined in `]-1,1[`.
`therefore` f(x) is differentiable in , ]1,1[.
(iii) Here `f(-1)` = log {`(-1)^(2) + 2}` - log 3 = 0
and `" "` f(1) = log `{(1)^(2) + 2}` - log 3 = 0
`therefore " " f(-1) = f(1)`
Thus the conditions of Rolle's theorem satisfies.
Let `" "` f'(c) = 0
`implies " " (2c)/(c^(2) + 2) = 0`
`implies " " c = 0 in ]-1 ,1[`
Hence Rolle's theorem verified .