Question

f(x) = x(x – 1)² in [0, 1]

Answer 1

We have, f(x) = x(x – 1)² in [0, 1]

Since, f(x) = x(x  – 1)² is a polynomial function it is continuous in [0,1] and differentiable in (0, 1)

Now, f(0) = 0 and f(1)

⇒ f(0) = f(1)

f satisfies the conditions of Rolle's theorem.

Hence, by Rolle's theorem there exists atleast one c ∈ (0, 1) such that f'(c) = 0

⇒ 3c² – 4c + 1 = 0

⇒ (3c – 1)(c – 1) = 0

⇒ c = `1/3 ∈ (0, 1)` 

Therefore, Rolle's theorem has been verified.