Verify Rolle's theorem for the function f(x) = cos2x in the interval `[0, pi]`.

Explanation

f(x) = cos 2x
Cosine function is continuous and differentiable at each point.
`therefore` f(x) is continous in the interval `[0,pi]` and differentiable in the interval `] 0 , pi [`.
Here `" "` f(0) = cos 0 = 1
and `" " f(pi) = cos 2 pi = 1`
`therefore " " f(0) = f(1)`
Thus the conditions of Rolle's theorem satisfies .
Now let f'(c) = 0
`implies " " -2 sin 2c = 0 " " [because f'(x) = -2 sin 2x]`
`implies " " sin 2c = 0`
`implies " " 2c = 0 , pi , 2pi....`
`implies " " c = 0 , pi//2 , pi....`
`c = pi//2 in ]0. pi[`
Hence Rolle's theorem verified .