The value of c in Rolle’s theorem for the function f(x) = x³ – 3x in the interval

Rolle’s Theorem states that, Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b), such that f(a) = f(b), where a and b are some real numbers.Then there exists some c in (a, b) such that f’(c) = 0.

We have, f(x) = x³ – 3x

Since, f(x) is a polynomial function it is continuous on and differentiable on

⇒

Now, as per Rolle’s Theorem, there exists at least one c ∈ , such that

f’(c) = 0

⇒ 3c² – 3 = 0 [∵ f’(x) = 3x² – 3 ]

⇒ c² = 1

⇒ c = ±1

⇒ c = 1 ∈