Verify Rolles theorem for the function `f(x)=sqrt(4-x^2)` on `[-2,\ 2]` .

Answer

N/a

Explanation

We have, `f(x) = sqrt(4-x^(2)) = (4-x^(2))^(1//2)`
(i) `f(x) = sqrt(4-x^(2))` is a continuous function.
[Since every polynomial function is a continuous function]
Hence, `f(x)` is continuous in `[-2,2]`.
(ii) `f'(x) = 1/2(4-x^(2))^(-1//2).(-2x)`
`= -x(1)/(sqrt(4-x^(2)))`. which exists everywhere except at `x = +- 2`.
Hence, ` f(x)` is differentiable in `(-2,2)`.
(iii) `f(-2) = sqrt((4-4)) = 0` and `f(2) = sqrt((4-4)) = 0`
`rArr " " f(-2) = f(2)`
conditions of Rolle's theorem are satisfied.
Hence, there exists a real number c such that `f'(c) = 0`.
`rArr -c(1)/(sqrt(4-c^(2))) = 0`
`rArr c = 0 in (-2,2)`
Hence, Rolle's thorem has been verified.