Question

Let ƒ(x) = (x – 3) (x – 4) (x – 5) (x – 6), then –

• A. ƒ ¢ (x) = 0 has four roots
• B. Three roots of ƒ ¢ (x) = 0 lie in (3, 4) Č (4, 5)Č (5, 6)
• C. The equation ƒ ¢ (x) = 0 has only one root
• D. Three roots of ƒ ¢ (x) = 0 lie in (2, 3) Č (3, 4)Č (4, 5)

Answer 1

Answer: (B)

Three roots of ƒ ¢ (x) = 0 lie in (3, 4) Č (4, 5)Č (5, 6)

Answer 2

ƒ(x) = (x – 3) (x – 4) (x – 5) (x – 6)

Ž ƒ(3) = ƒ(4) = ƒ(5) = ƒ(6) = 0

\ by Rolle’s theorem, there exist

₁ Ī (3, 4), a₂ Ī (4, 5) and a₃ Ī (5, 6) such that

Ģ (a_i) = 0, i = 1, 2, 3.

Since ƒ¢(x) is a cubic polynomial therefore a₁, a₂, a₃ are the only roots of ƒ ¢ (x) = 0.