583. Verify Rolle's Theorem for the function f(x) = x(x - 1)² in [0, 1]. 

Here f(x) = x(x - 1)² = x³ - 2 x² + x
It is a polynomial in x.
(i) Since every polynomial in x is a continuous function for every value of x.
∴ f(x) is continuous in [0, 1].
(ii) f'(x) = 3 x² - 4 x + 1, which existsyn (0, 1)
∴ f(x) is derivable in (0, 1).
(iii) f(0) = 0, f(1) = 0
∴ f(0) = f(1)
∴ 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 1.