596. Verify Rolle's Theorem for the function f(x) = sin² x in [0, ].

Here f(x) = sin² x
(i) We know that sin x is continuous in [0, ]
Now sin²x, being the product of two continuous function sin x and sin x, is continuous in [0, ].
(ii) f'(x) = 2 sin x cos x, which exists in (0, )
∴ f(x) is derivable in (0, π).
(iii) f(0) = sin² 0 = 0, f() = sin² π = 0
∴ f(0) = f()
∴ f{x) satisfies all the conditions of Rolle's Theorem
∴ there must exist at least one value c of x such that
f'(c) = 0 where 0 < c < .