Question

The equation x³− 3x + [a] = 0, where [.] denotes the greatest integer function, will have real and distinct roots if

• A. a ∈ (-∞, 2)
• B. a ∈ (0, 2)
• C. a ∈ (∞, - 2) ∪ (0, ∞)
• D. a ∈[-1, 2)

Answer 1

Answer: (D)

a ∈[-1, 2)

Answer 2

Let f(x) = x³ − 3x + 1 ⇒ f(x) = 3(x – 1)(x + 1).

Clearly x = −1 and x = 1 are the points of local maxima and local minima respectively for y = f(x). Now f(x) = 0 will have real and distinct roots if f(1).f(-1) < 0 ⇒ ([a] + 2) ([a] - 2) < 0

⇒ −2 < [a] < 2

⇒ a ∈ [-1,2)