Find the number 'c' that satisfies the conclusion of Rolle's theorem for the function f(x) = (x² -1)(x-2) in (1, 2]?

To find 'c', solve f'(x)=0

⇒ (x² − 1) × 1 + 2x(x − 2) = 0

⇒ x²− 1 + 2x² − 4x = 0

⇒ 3x² − 4x−1 = 0

It's a quadratic equation so, solve it using quadratic formula

⇒ [4±√16-(4)(3)(-1)] / (2(3))

⇒x = (2±√7)/(3)

⇒x = (2+√7)/(3) and x = (2-√7)/(3)

x = (2+ √7)/(3) = 1.55 ∈(1,2]

x = (2-√7)/(3) = -.215 ∉(1,2]

Hence, c= (2+√7)/(3) satisfies the conclusion of Rolle's theorem for the given function