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

Explanation

f(x) = `x^(2)`
(i) f(x) is defined in the interval `[-1,1]` .
`therefore` f(x) is continous function in `[-1,1]`.
(ii) f'(x) = 2x
Which is defined in `]-1,1[.`
`therefore` f(x) is differentiable in ]`-1, 1[` .
(iii) Here f(-1) = `(-1)^(2) = 1`
and `" " f(1) = (1)^(2) = 1`
`therefore " " f(-1) = f(1)`
Thus the conditions of Roll's theorem satisfies .
Now let f'(c) = 0
`implies " " 2c = 0`
`implies " " c = 0 in ]-1 , 1[`.
Hence Roll's theorem verified .